Geometric Gems: An Appreciation For Geometric Curiosities - Volume I: The Wonders Of Triangles (Problem Solving in Mathematics and Beyond)

Contents
Acknowledgments
About the Authors
Introduction
Geometric Curiosities: Introducing the Triangle
	Curiosity 1. Angles at the Incenter
	Curiosity 2. An Unexpected Perpendicularity
	Curiosity 3. A Most Unusual Line Bisection
	Curiosity 4. Gergonne’s Discovery Involving the Inscribed Circle of a Triangle
	Curiosity 5. Another Unexpected Concurrency for Gergonne’s Triangle
	Curiosity 6. A Novelty Concerning the Circumscribed and Inscribed Circles
	Curiosity 7. Concurrency Involving the Inscribed Circle
	Curiosity 8. A More General Concurrency Involving the Inscribed Circle
	Curiosity 9. An Inscribed Circle of a General Triangle Generates Equal Angles
	Curiosity 10. Surprising Equal Angles
	Curiosity 11. An Inscribed Circle of a Triangle Generates its Circumscribed Circle
	Curiosity 12. The Feet of the Altitudes Partition a Triangle into Three Pairs of Equal-Area Triangles
	Curiosity 13. The Noteworthy Position of the Orthocenter of a Triangle
	Curiosity 14. A Circle Intersects a Triangle to Generate Lots of Equal Segments
	Curiosity 15. A Strange Appearance of a Congruent Triangle
	Curiosity 16. The Orthocenter Joins Three Other Points in Collinearity
	Curiosity 17. The Orthocenter Appears as the Midpoint of a Line Segment
	Curiosity 18. A Conglomeration of Perpendiculars Generates Equal Line Segments
	Curiosity 19. Yet Another Unexpected Collinearity
	Curiosity 20. Introducing the Orthic Triangle
	Curiosity 21. Finding the Perimeter of the Orthic Triangle
	Curiosity 22. The Orthic Triangle is a Triangle’s Smallest Inscribed Triangle
	Curiosity 23. The Orthic Triangle’s Surprising Similar Partner
	Curiosity 24. A Nice Concurrency Generated by the Orthic Triangle
	Curiosity 25. Orthic Triangle Generates an Isosceles Triangle
	Curiosity 26. The Orthic Triangle Generates a Parallelogram
	Curiosity 27. Concyclic Points Generated by the Orthic Triangle
	Curiosity 28. The Orthic Triangle Generates Concurrent Lines
	Curiosity 29. The Orthic Triangle Generates Collinear Points
	Curiosity 30. An Unexpected Property of Altitude Feet of a Triangle
	Curiosity 31. How a Non-Isosceles Triangle Can Generate a Parallelogram
	Curiosity 32. Perpendiculars Generating Unexpected Parallel Lines
	Curiosity 33. Inscribed Triangle Generates Collinearity
	Curiosity 34. Unexpected Congruent Triangles in a Circle
	Curiosity 35. An Unexpected Product Equality
	Curiosity 36. An Unusual Product of Two Sides of a Triangle
	Curiosity 37. Simson’s Theorem
	Curiosity 38. An Extension of Simson’s Theorem
	Curiosity 39. An Interesting Aspect of Simson’s Theorem
	Curiosity 40. A Parallel to the Simson Line
	Curiosity 41. Two Triangles Related by a Common Point: Circumcenter – Centroid
	Curiosity 42. Introducing the Medians of a Triangle with Some of Their Amazing Properties
	Curiosity 43. The Median of a Triangle is Equidistant from Two Vertices
	Curiosity 44. The Special Median of a Right Triangle
	Curiosity 45. Medians Partition Any Triangle into Four Congruent Triangles
	Curiosity 46. How the Centroid Helps Create a Similar Triangle
	Curiosity 47. The Centroid can Provide a Most Unusual Balance
	Curiosity 48. Distances from a Triangle’s Vertices to a Random Line
	Curiosity 49. A Special Centroid Property When Two Medians are Perpendicular
	Curiosity 50. The Centroid’s Amazing Property
	Curiosity 51. Some Median Surprises
	Curiosity 52. More Median Marvels
	Curiosity 53. Comparing Medians to Triangle Perimeters
	Curiosity 54. Median Extensions Generate Collinearity
	Curiosity 55. Two Unusual Triangles Share a Common Centroid
	Curiosity 56. The Circumscribed Circle Revisited: The Incredible Relationship of the Centers of the Circumscribed Circles of the Median Triangles
	Curiosity 57. An Astonishing Equality
	Curiosity 58. An Unexpected Simultaneous Bisection and Quadrisection in a Triangle
	Curiosity 59. Noteworthy Triangle Area Relations
	Curiosity 60. A Special Feature of a Random Point in an Equilateral Triangle
	Curiosity 61. A Special Feature of a Point Outside of an Equilateral Triangle
	Curiosity 62. Using an Equilateral Triangle to Trisect a Line Segment
	Curiosity 63. Another Way to Trisect a Line Segment with an Equilateral Triangle
	Curiosity 64. An Unusual Construction of a 30°-60°-90° Triangle Inside an Equilateral Triangle
	Curiosity 65. An Unusual Product of Segments in an Equilateral Triangle
	Curiosity 66. A Surprising Relationship Between an Isosceles Triangle and an Equilateral Triangle
	Curiosity 67. An Unexpected Equality in an Isosceles Triangle
	Curiosity 68. Another Unexpected Equality in an Isosceles Triangle
	Curiosity 69. A Remarkable Geometric Equality
	Curiosity 70. Another Remarkable Geometric Equality
	Curiosity 71. An Unexpected Perpendicularity
	Curiosity 72. The Unexpected Property of an Altitude to the Side of an Isosceles Triangle
	Curiosity 73. Peculiar Property of Isosceles Triangles
	Curiosity 74. A Bizarre Connection: The Triangle Incenter on its Circumcircle
	Curiosity 75. Collinear Points Generate an Angle Bisector
	Curiosity 76. Unexpected Similar Triangles
	Curiosity 77. Perpendiculars to Four Angle Bisectors of a Triangle Reveal Four Collinear Points
	Curiosity 78. A Convoluted Bisection of a Line Segment
	Curiosity 79. Determining the Perimeter of a Triangle Without Measuring its Side Lengths
	Curiosity 80. A Strange Angle Trisection
	Curiosity 81. A Most Unexpected Equality
	Curiosity 82. A Triangle Peculiarity
	Curiosity 83. A Counterintuitive Area Equality of Triangles
	Curiosity 84. Parallel Lines Create a Double Area Triangle
	Curiosity 85. Unexpected Triangle Area Relationships
	Curiosity 86. Creating a Triangle Whose Area is Three-Quarters the Area of a Given Triangle
	Curiosity 87. An Astounding Construction: Similar Triangles Whose Area Ratio is 1:4
	Curiosity 88. Unforeseen Equality of Inscribed Triangles
	Curiosity 89. The Unanticipated Commonality of Equal Area Triangles
	Curiosity 90. How a Random Point Divides the Area of a Triangle in Half
	Curiosity 91. Determining a Triangle One-Third of the Area of a Given Triangle
	Curiosity 92. Trisection Points Partitioning a Triangle
	Curiosity 93. Further Surprises Provided by the Trisection Points of Triangle Sides
	Curiosity 94. Another Surprise Provided by the Trisection Points of Triangle Sides
	Curiosity 95. Yet Another Surprise Provided by the Trisection Points of Triangle Sides
	Curiosity 96. Medians and Trisectors Partitioning a Triangle with an Unexpected Result
	Curiosity 97. The Unforeseen Characteristic of a Random Triangle with a 60° Angle
	Curiosity 98. Another Surprising Feature of a Random Triangle with a 60° Angle
	Curiosity 99. Another Unexpected Collinearity
	Curiosity 100. Reflections of Triangles Generate Concurrent Circles and Concurrent Lines
	Curiosity 101. The Wonders of Three Concurrent Congruent Circles
	Curiosity 102. Further Unexpected Concurrencies
	Curiosity 103. One Concurrency Generates Another Concurrency
	Curiosity 104. Unusual Perpendiculars Generating Concurrencies
	Curiosity 105. A Most Unexpected Concurrency
	Curiosity 106. An Intriguing Concurrency
	Curiosity 107. Another Unexpected and Unforeseen Concurrency
	Curiosity 108. A Most Unexpected Concurrency from a Triangle
	Curiosity 109. Astounding Point Property in a Triangle
	Curiosity 110. A Counterintuitive Concurrency
	Curiosity 111. Concurrent Angle Bisectors of Triangles with a Common Base
	Curiosity 112. An Unexpected Concurrency with the Circumscribed Circle
	Curiosity 113. Another Surprising Aspect of Concurrent Cevians
	Curiosity 114. A Surprising Equality
	Curiosity 115. An Unforeseen Triangle Surprise
	Curiosity 116. A Remarkable Property of Two Triangles with a Common Base
	Curiosity 117. A Surprising Line Partitioning
	Curiosity 118. The Hidden Length Equality of Line Segments in a Triangle
	Curiosity 119. The Unexpected Angle Measure
	Curiosity 120. An Unexpected Equality from a Right Triangle
	Curiosity 121. The Conundrum: Perpendicular or Parallel
	Curiosity 122. Multiple Midpoints in a Right Triangle Determine Equal Line Segments
	Curiosity 123. Perpendiculars in Right Triangles that Generate Equal Angles
	Curiosity 124. Right Triangles Sharing a Common Hypotenuse Generate an Unexpected Equality
	Curiosity 125. The Pythagorean Theorem Revisited Geometrically
	Curiosity 126. Squares on Triangle Sides Produce Noteworthy Surprises
	Curiosity 127. More Properties Generated by Squares on Triangle Sides
	Curiosity 128. An Unexpected Perpendicularity
	Curiosity 129. A Surprising Concurrency from Squares on the Sides of a Triangle
	Curiosity 130. Another Surprising Concurrency from Squares on Triangle Sides
	Curiosity 131. Using Squares on Triangle Sides to Create another Surprising Concurrency
	Curiosity 132. Squares on the Legs of a Right Triangle Produce an Unexpected Equality
	Curiosity 133. More About Squares on the Legs of a Right Triangle
	Curiosity 134. More Placements of Squares on Right Triangles
	Curiosity 135. Another Unexpected Area Equality
	Curiosity 136. The Square on the Hypotenuse of a Right Triangle
	Curiosity 137. A Truly Unexpected Collinearity
	Curiosity 138. A Most Unusual Procedure to Divide a Square into Two Equal Parts
	Curiosity 139. Doubling a Square
	Curiosity 140. A Strange Construction of Parallel Lines
	Curiosity 141. An Unusual Technique to Find the Midpoint of a Line Segment
	Curiosity 142. The Unexpected Appearance of Parallel Lines
	Curiosity 143. The Unanticipated Parallel Line
	Curiosity 144. The Surprising Perpendicularity
	Curiosity 145. Another Unexpected Perpendicularity
	Curiosity 146. Yet Another Unexpected Right Angle
	Curiosity 147. Four Important Concyclic Points
	Curiosity 148. Four Remarkable Concyclic Points
	Curiosity 149. Four Unexpected Concyclic Points
	Curiosity 150. Perpendiculars that Generate Concyclic Points
	Curiosity 151. Altitudes and Circles that Generate Another Circle
	Curiosity 152. More Unexpected Concyclic Points
	Curiosity 153. A Surprising Five-Point Circle
	Curiosity 154. The Famous Nine-Point Circle
	Curiosity 155. A Collinearity with the Center of the Nine-Point Circle
	Curiosity 156. The Meeting of the Three Famous Triangle Centers
	Curiosity 157. Properties of the Nine-Point Circle
	Curiosity 158. More Properties of the Nine-Point Circle
	Curiosity 159. An Unexpected Collinearity
	Curiosity 160. A Concurrency Generated by the Orthic Triangle
	Curiosity 161. Altitudes Produce a Concurrency and Equality
	Curiosity 162. Napoleon’s Contribution to Mathematics
	Curiosity 163. Napoleon’s Minimum Distance Point (the Fermat Point)
	Curiosity 164. When the Minimum Distance Point is not Inside the Triangle
	Curiosity 165. Extensions of Napoleon’s Theorem
	Curiosity 166. Overlapping Side-Equilateral Triangles
	Curiosity 167. Surprising Triangle Area Relationship
	Curiosity 168. The Centroid Enters the Previous Configuration
	Curiosity 169. The Emergence of Another Equilateral Triangle
	Curiosity 170. A Novel Way of Finding the Center of the Circumscribed Circle
	Curiosity 171. A Concurrency Point of Circles
	Curiosity 172. The Famous Miquel Theorem
	Curiosity 173. Miquel’s Similar Triangles
	Curiosity 174. The Astounding Morley’s Theorem
	Curiosity 175. Morley’s Theorem Extended
Proofs of the Triangle Curiosities
	Curiosity 1. Angles at the Incenter
	Curiosity 2. An Unexpected Perpendicularity
	Curiosity 3. A Most Unusual Line Bisection
	Curiosity 4. Gergonne’s Discovery Involving the Inscribed Circle of a Triangle
	Curiosity 5. Another Unexpected Concurrency for Gergonne’s Triangle
	Curiosity 6. A Novelty Concerning the Circumscribed and Inscribed Circles
	Curiosity 7. Concurrency Involving the Inscribed Circle
	Curiosity 8. A More General Concurrency Involving the Inscribed Circle
	Curiosity 9. An Inscribed Circle of a General Triangle Generates Equal Angles
	Curiosity 10. Surprising Equal Angles
	Curiosity 11. An Inscribed Circle of a Triangle Generates its Circumscribed Circle
	Curiosity 12. The Feet of the Altitudes Partition a Triangle into Three Pairs of Equal-Area Triangles
	Curiosity 13. The Noteworthy Position of the Orthocenter of a Triangle
	Curiosity 14. A Circle Intersects a Triangle to Generate Lots of Equal Segments
	Curiosity 15. A Strange Appearance of a Congruent Triangle
	Curiosity 16. The Orthocenter Joins Three Other Points in Collinearity
	Curiosity 17. The Orthocenter Appears as the Midpoint of a Line Segment
	Curiosity 18. A Conglomeration of Perpendiculars Generates Equal Line Segments
	Curiosity 19. Yet Another Unexpected Collinearity
	Curiosity 20. Introducing the Orthic Triangle
	Curiosity 21. Finding the Perimeter of the Orthic Triangle
	Curiosity 22. The Orthic Triangle is a Triangle’s Smallest Inscribed Triangle
	Curiosity 23. The Orthic Triangle’s Surprising Similar Partner
	Curiosity 24. A Nice Concurrency Generated by the Orthic Triangle
	Curiosity 25. Orthic Triangle Generates an Isosceles Triangle
	Curiosity 26. The Orthic Triangle Generates a Parallelogram
	Curiosity 27. Concyclic Points Generated by the Orthic Triangle
	Curiosity 28. The Orthic Triangle Generates Concurrent Lines
	Curiosity 29. The Orthic Triangle Generates Collinear Points
	Curiosity 30. An Unexpected Property of Altitude Feet of a Triangle
	Curiosity 31. How a Non-Isosceles Triangle Can Generate a Parallelogram
	Curiosity 32. Perpendiculars Generating Unexpected Parallel Lines
	Curiosity 33. Tangents and Inscribed-Triangle Sides Generate a Surprising Collinearity
	Curiosity 34. Creating Congruent Triangles Inscribed in the Same Circle
	Curiosity 35. An Unexpected Product Equality
	Curiosity 36. An Unusual Product of Two Sides of a Triangle
	Curiosity 37. Simson’s Theorem
	Curiosity 38. An Extension of Simson’s Theorem
	Curiosity 39. An Interesting Aspect of Simson’s Theorem
	Curiosity 40. A Parallel to the Simson Line
	Curiosity 41. Two Triangles Related by a Common Point: Circumcenter – Centroid
	Curiosity 42. Introducing the Medians of a Triangle with Some of Their Amazing Properties
	Curiosity 43. The Median of a Triangle is Equidistant from Two Vertices
	Curiosity 44. The Special Median of a Right Triangle
	Curiosity 45. Medians Partition Any Triangle into Four Congruent Triangles
	Curiosity 46. How the Centroid Helps Create a Similar Triangle
	Curiosity 47. The Centroid can Provide a Most Unusual Balance
	Curiosity 48. Distances from a Triangle’s Vertices to a Random Line
	Curiosity 49. A Special Centroid Property When Two Medians are Perpendicular
	Curiosity 50. The Centroid’s Amazing Property
	Curiosity 51. Some Median Surprises
	Curiosity 52. More Median Marvels
	Curiosity 53. Comparing Medians to Triangle Perimeters
	Curiosity 54. Median Extensions Generate Collinearity
	Curiosity 55. Two Unusual Triangles Share a Common Centroid
	Curiosity 56. The Circumscribed Circle Revisited: The Incredible Relationship of the Centers of the Circumscribed Circles of the Median Triangles
	Curiosity 57. An Astonishing Equality
	Curiosity 58. An Unexpected Simultaneous Bisection and Quadrisection in a Triangle
	Curiosity 59. Noteworthy Triangle Area Relations
	Curiosity 60. A Special Feature of a Random Point in an Equilateral Triangle
	Curiosity 61. A Special Feature of a Point Outside of an Equilateral Triangle
	Curiosity 62. Using an Equilateral Triangle to Trisect a Line Segment
	Curiosity 63. Another Way to Trisect a Line Segment with an Equilateral Triangle
	Curiosity 64. An Unusual Construction of a 30°-60°- 90° Triangle inside an Equilateral Triangle
	Curiosity 65. An Unusual Product of Segments in an Equilateral Triangle
	Curiosity 66. A Surprising Relationship between an Isosceles Triangle and an Equilateral Triangle
	Curiosity 67. An Unexpected Equality in an Isosceles Triangle
	Curiosity 68. Another Unexpected Equality in an Isosceles Triangle
	Curiosity 69. A Remarkable Geometric Equality
	Curiosity 70. Another Remarkable Geometric Equality
	Curiosity 71. An Unexpected Perpendicularity
	Curiosity 72. The Unexpected Property of an Altitude to the Side of an Isosceles Triangle
	Curiosity 73. Peculiar Property of Isosceles Triangles
	Curiosity 74. A Bizarre Connection: The Triangle Incenter on its Circumcircle
	Curiosity 75. Collinear Points Generate an Angle Bisector
	Curiosity 76. Unexpected Similar Triangles
	Curiosity 77. Perpendiculars to Four Angle Bisectors of a Triangle Reveal Four Collinear Points
	Curiosity 78. A Convoluted Bisection of a Line Segment
	Curiosity 79. Determining the Perimeter of a Triangle Without Measuring its Side Lengths
	Curiosity 80. A Strange Angle Trisection
	Curiosity 81. A Most Unexpected Equality
	Curiosity 82. A Triangle Peculiarity
	Curiosity 83. A Counterintuitive Area Equality of Triangles
	Curiosity 84. Parallel Lines Create a Double Area Triangle
	Curiosity 85. Unexpected Triangle Area Relationships
	Curiosity 86. Creating a Triangle Whose Area is Three-Quarters the Area of a Given Triangle
	Curiosity 87. An Astounding Construction: Similar Triangles Whose Area Ratio is 1:4
	Curiosity 88. Unforeseen Equality of Inscribed Triangles
	Curiosity 89. The Unanticipated Commonality of Equal Area Triangles
	Curiosity 90. How a Random Point Divides the Area of a Triangle in Half
	Curiosity 91. Determining a Triangle One-Third of the Area of a Given Triangle
	Curiosity 92. Trisection Points Partitioning a Triangle
	Curiosity 93. Further Surprises Provided by the Trisection Points of Triangle Sides
	Curiosity 94. Another Surprise Provided by the Trisection Points of Triangle Sides
	Curiosity 95. Yet Another Surprise Provided by the Trisection Points of Triangle Sides
	Curiosity 96. Medians and Trisectors Partitioning a Triangle with an Unexpected Result
	Curiosity 97. The Unforeseen Characteristic of a Random Triangle with a 60° Angle
	Curiosity 98. Another Surprising Feature of a Random Triangle with a 60° Angle
	Curiosity 99. Another Unexpected Collinearity
	Curiosity 100. Reflections of Triangles Generate Concurrent Circles and Concurrent Lines
	Curiosity 101. The Wonders of Three Concurrent Congruent Circles
	Curiosity 102. Further Unexpected Concurrencies
	Curiosity 103. One Concurrency Generates Another Concurrency
	Curiosity 104. Unusual Perpendiculars Generating Concurrencies
	Curiosity 105. A Most Unexpected Concurrency
	Curiosity 106. An Intriguing Concurrency
	Curiosity 107. Another Unexpected and Unforeseen Concurrency
	Curiosity 108. A Most Unexpected Concurrency from a Triangle
	Curiosity 109. Astounding Point Property in a Triangle
	Curiosity 110. A Counterintuitive Concurrency
	Curiosity 111. Concurrent Angle Bisectors of Triangles with a Common Base
	Curiosity 112. An Unexpected Concurrency with the Circumscribed Circle
	Curiosity 113. Another Surprising Aspect of Concurrent Cevians
	Curiosity 114. A Surprising Equality
	Curiosity 115. An Unforeseen Triangle Surprise
	Curiosity 116. A Remarkable Property of Two Triangles with a Common Base
	Curiosity 117. A Surprising Line Partitioning
	Curiosity 118. The Hidden Length Equality of Line Segments in a Triangle
	Curiosity 119. The Unexpected Angle Measure
	Curiosity 120. An Unexpected Equality from a Right Triangle
	Curiosity 121. The Conundrum: Perpendicular or Parallel
	Curiosity 122. Multiple Midpoints in a Right Triangle Determine Equal Line Segments
	Curiosity 123. Perpendiculars in Right Triangles that Generate Equal Angles
	Curiosity 124. Right Triangles Sharing a Common Hypotenuse Generate an Unexpected Equality
	Curiosity 125. The Pythagorean Theorem Revisited Geometrically
	Curiosity 126. Squares on Triangle Sides Produce Noteworthy Surprises
	Curiosity 127. More Properties Generated by Squares on Triangle Sides
	Curiosity 128. An Unexpected Perpendicularity
	Curiosity 129. A Surprising Concurrency from Squares on the Sides of a Triangle
	Curiosity 130. Another Surprising Concurrency from Squares on Triangle Sides
	Curiosity 131. Using Squares on Triangle Sides to Create another Surprising Concurrency
	Curiosity 132. Squares on the Legs of a Right Triangle Produce an Unexpected Equality
	Curiosity 133. More About Squares on the Legs of a Right Triangle
	Curiosity 134. More Placements of Squares on Right Triangles
	Curiosity 135. Another Unexpected Area Equality
	Curiosity 136. The Square on the Hypotenuse of a Right Triangle
	Curiosity 137. A Truly Unexpected Collinearity
	Curiosity 138. A Most Unusual Procedure to Divide a Square into Two Equal Parts
	Curiosity 139. Doubling a Square
	Curiosity 140. A Strange Construction of Parallel Lines
	Curiosity 141. An Unusual Technique to Find the Midpoint of a Line Segment
	Curiosity 142. The Unexpected Appearance of Parallel Lines
	Curiosity 143. The Unanticipated Parallel Line
	Curiosity 144. The Surprising Perpendicularity
	Curiosity 145. Another Unexpected Perpendicularity
	Curiosity 146. Yet Another Unexpected Right Angle
	Curiosity 147. Four Important Concyclic Points
	Curiosity 148. Four Remarkable Concyclic Points
	Curiosity 149. Four Unexpected Concyclic Points
	Curiosity 150. Perpendiculars that Generate Concyclic Points
	Curiosity 151. Altitudes and Circles that Generate Another Circle
	Curiosity 152. More Unexpected Concyclic Points
	Curiosity 153. A Surprising Five-Point Circle
	Curiosity 154. The Famous Nine-Point Circle
	Curiosity 155. A Collinearity with the Center of the Nine-Point Circle
	Curiosity 156. The Meeting of the Three Famous Triangle Centers
	Curiosity 157. Properties of the Nine-Point Circle
	Curiosity 158. More Properties of the Nine-Point Circle
	Curiosity 159. An Unexpected Collinearity
	Curiosity 160. A Concurrency Generated by the Orthic Triangle
	Curiosity 161. Altitudes Produce a Concurrency and Equality
	Curiosity 162. Napoleon’s Contribution to Mathematics
	Curiosity 163. Napoleon’s Minimum Distance Point (The Fermat Point)
	Curiosity 164. When the Minimum Distance Point is not Inside the Triangle
	Curiosity 165. Extensions of Napoleon’s Theorem
	Curiosity 166. Overlapping Side-Equilateral Triangles
	Curiosity 167. Surprising Triangle Area Relationship
	Curiosity 168. The Centroid Enters the Previous Configuration
	Curiosity 169. The Emergence of Another Equilateral Triangle
	Curiosity 170. A Novel Way of Finding the Center of the Circumscribed Circle
	Curiosity 171. A Concurrency Point of Circles
	Curiosity 172. The Famous Miquel Theorem
	Curiosity 173. Miquel’s Similar Triangles
	Curiosity 174. The Astounding Morley’s Theorem
	Curiosity 175. Morley’s Theorem Extended
Toolbox
	Introduction: The Geometry Toolbox
	A.	Tools You Are Probably Familiar with from the High School Geometry Course
		A1: Congruence of Triangles
		A2: Similarity of Triangles
		A3: Right Triangle Properties (See Figure A3)
		A4: Angles Related to a Circle (see Figure A4)
		A5:	Tangents, Secants, and Chords: Segments of a Circle (see Figure A5)
		A6: The Law of Sines and the Law of Cosines
		A7: Angle Sum and Difference Identities
	B.	Less Familiar Tools—However, Useful and Fascinating
		B1. Interior Angle Bisector in a Triangle
		B2. Exterior Angle Bisector in a Triangle
		B3. Menelaus’ Theorem
		B4. Ceva’s Theorem
		B5. The Trigonometric Version of Ceva’s Theorem
		B6. Ceva’s Theorem Extended
		B7. Desargues’s Theorem
		B8. Stewart’s Theorem
		B9. Theorem of Apollonius
		B10. Ptolemy’s Theorem
		B11. Isometries: Reflection and Rotation
		B12. Homothety and Similarity
		B13. Polarity on Circles
Index
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	Contents
	Acknowledgments
	About the Authors
		Introduction
		Geometric Curiosities: Introducing the Triangle
			Curiosity 1. Angles at the Incenter
				Curiosity 2. An Unexpected Perpendicularity
				Curiosity 3. A Most Unusual Line Bisection
				Curiosity 4. Gergonne’s Discovery Involving the Inscribed Circle of a Triangle
				Curiosity 5. Another Unexpected Concurrency for Gergonne’s Triangle
				Curiosity 6. A Novelty Concerning the Circumscribed and Inscribed Circles
				Curiosity 7. Concurrency Involving the Inscribed Circle
				Curiosity 8. A More General Concurrency Involving the Inscribed Circle
				Curiosity 9. An Inscribed Circle of a General Triangle Generates Equal Angles
				Curiosity 10. Surprising Equal Angles
				Curiosity 11. An Inscribed Circle of a Triangle Generates its Circumscribed Circle
				Curiosity 12. The Feet of the Altitudes Partition a Triangle into Three Pairs of Equal-Area Triangles
				Curiosity 13. The Noteworthy Position of the Orthocenter of a Triangle
				Curiosity 14. A Circle Intersects a Triangle to Generate Lots of Equal Segments
				Curiosity 15. A Strange Appearance of a Congruent Triangle
				Curiosity 16. The Orthocenter Joins Three Other Points in Collinearity
				Curiosity 17. The Orthocenter Appears as the Midpoint of a Line Segment
				Curiosity 18. A Conglomeration of Perpendiculars Generates Equal Line Segments
				Curiosity 19. Yet Another Unexpected Collinearity
				Curiosity 20. Introducing the Orthic Triangle
				Curiosity 21. Finding the Perimeter of the Orthic Triangle
				Curiosity 22. The Orthic Triangle is a Triangle’s Smallest Inscribed Triangle
				Curiosity 23. The Orthic Triangle’s Surprising Similar Partner
				Curiosity 24. A Nice Concurrency Generated by the Orthic Triangle
				Curiosity 25. Orthic Triangle Generates an Isosceles Triangle
				Curiosity 26. The Orthic Triangle Generates a Parallelogram
				Curiosity 27. Concyclic Points Generated by the Orthic Triangle
				Curiosity 28. The Orthic Triangle Generates Concurrent Lines
				Curiosity 29. The Orthic Triangle Generates Collinear Points
				Curiosity 30. An Unexpected Property of Altitude Feet of a Triangle
				Curiosity 31. How a Non-Isosceles Triangle Can Generate a Parallelogram
				Curiosity 32. Perpendiculars Generating Unexpected Parallel Lines
				Curiosity 33. Inscribed Triangle Generates Collinearity
				Curiosity 34. Unexpected Congruent Triangles in a Circle
				Curiosity 35. An Unexpected Product Equality
				Curiosity 36. An Unusual Product of Two Sides of a Triangle
				Curiosity 37. Simson’s Theorem
				Curiosity 38. An Extension of Simson’s Theorem
				Curiosity 39. An Interesting Aspect of Simson’s Theorem
				Curiosity 40. A Parallel to the Simson Line
				Curiosity 41. Two Triangles Related by a Common Point: Circumcenter – Centroid
				Curiosity 42. Introducing the Medians of a Triangle with Some of Their Amazing Properties
				Curiosity 43. The Median of a Triangle is Equidistant from Two Vertices
				Curiosity 44. The Special Median of a Right Triangle
				Curiosity 45. Medians Partition Any Triangle into Four Congruent Triangles
				Curiosity 46. How the Centroid Helps Create a Similar Triangle
				Curiosity 47. The Centroid can Provide a Most Unusual Balance
				Curiosity 48. Distances from a Triangle’s Vertices to a Random Line
				Curiosity 49. A Special Centroid Property When Two Medians are Perpendicular
				Curiosity 50. The Centroid’s Amazing Property
				Curiosity 51. Some Median Surprises
				Curiosity 52. More Median Marvels
				Curiosity 53. Comparing Medians to Triangle Perimeters
				Curiosity 54. Median Extensions Generate Collinearity
				Curiosity 55. Two Unusual Triangles Share a Common Centroid
				Curiosity 56. The Circumscribed Circle Revisited: The Incredible Relationship of the Centers of the Circumscribed Circles of the Median Triangles
				Curiosity 57. An Astonishing Equality
				Curiosity 58. An Unexpected Simultaneous Bisection and Quadrisection in a Triangle
				Curiosity 59. Noteworthy Triangle Area Relations
				Curiosity 60. A Special Feature of a Random Point in an Equilateral Triangle
				Curiosity 61. A Special Feature of a Point Outside of an Equilateral Triangle
				Curiosity 62. Using an Equilateral Triangle to Trisect a Line Segment
				Curiosity 63. Another Way to Trisect a Line Segment with an Equilateral Triangle
				Curiosity 64. An Unusual Construction of a 30°-60°-90° Triangle Inside an Equilateral Triangle
				Curiosity 65. An Unusual Product of Segments in an Equilateral Triangle
				Curiosity 66. A Surprising Relationship Between an Isosceles Triangle and an Equilateral Triangle
				Curiosity 67. An Unexpected Equality in an Isosceles Triangle
				Curiosity 68. Another Unexpected Equality in an Isosceles Triangle
				Curiosity 69. A Remarkable Geometric Equality
				Curiosity 70. Another Remarkable Geometric Equality
				Curiosity 71. An Unexpected Perpendicularity
				Curiosity 72. The Unexpected Property of an Altitude to the Side of an Isosceles Triangle
				Curiosity 73. Peculiar Property of Isosceles Triangles
				Curiosity 74. A Bizarre Connection: The Triangle Incenter on its Circumcircle
				Curiosity 75. Collinear Points Generate an Angle Bisector
				Curiosity 76. Unexpected Similar Triangles
				Curiosity 77. Perpendiculars to Four Angle Bisectors of a Triangle Reveal Four Collinear Points
				Curiosity 78. A Convoluted Bisection of a Line Segment
				Curiosity 79. Determining the Perimeter of a Triangle Without Measuring its Side Lengths
				Curiosity 80. A Strange Angle Trisection
				Curiosity 81. A Most Unexpected Equality
				Curiosity 82. A Triangle Peculiarity
				Curiosity 83. A Counterintuitive Area Equality of Triangles
				Curiosity 84. Parallel Lines Create a Double Area Triangle
				Curiosity 85. Unexpected Triangle Area Relationships
				Curiosity 86. Creating a Triangle Whose Area is Three-Quarters the Area of a Given Triangle
				Curiosity 87. An Astounding Construction: Similar Triangles Whose Area Ratio is 1:4
				Curiosity 88. Unforeseen Equality of Inscribed Triangles
				Curiosity 89. The Unanticipated Commonality of Equal Area Triangles
				Curiosity 90. How a Random Point Divides the Area of a Triangle in Half
				Curiosity 91. Determining a Triangle One-Third of the Area of a Given Triangle
				Curiosity 92. Trisection Points Partitioning a Triangle
				Curiosity 93. Further Surprises Provided by the Trisection Points of Triangle Sides
				Curiosity 94. Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 95. Yet Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 96. Medians and Trisectors Partitioning a Triangle with an Unexpected Result
				Curiosity 97. The Unforeseen Characteristic of a Random Triangle with a 60° Angle
				Curiosity 98. Another Surprising Feature of a Random Triangle with a 60° Angle
				Curiosity 99. Another Unexpected Collinearity
				Curiosity 100. Reflections of Triangles Generate Concurrent Circles and Concurrent Lines
				Curiosity 101. The Wonders of Three Concurrent Congruent Circles
				Curiosity 102. Further Unexpected Concurrencies
				Curiosity 103. One Concurrency Generates Another Concurrency
				Curiosity 104. Unusual Perpendiculars Generating Concurrencies
				Curiosity 105. A Most Unexpected Concurrency
				Curiosity 106. An Intriguing Concurrency
				Curiosity 107. Another Unexpected and Unforeseen Concurrency
				Curiosity 108. A Most Unexpected Concurrency from a Triangle
				Curiosity 109. Astounding Point Property in a Triangle
				Curiosity 110. A Counterintuitive Concurrency
				Curiosity 111. Concurrent Angle Bisectors of Triangles with a Common Base
				Curiosity 112. An Unexpected Concurrency with the Circumscribed Circle
				Curiosity 113. Another Surprising Aspect of Concurrent Cevians
				Curiosity 114. A Surprising Equality
				Curiosity 115. An Unforeseen Triangle Surprise
				Curiosity 116. A Remarkable Property of Two Triangles with a Common Base
				Curiosity 117. A Surprising Line Partitioning
				Curiosity 118. The Hidden Length Equality of Line Segments in a Triangle
				Curiosity 119. The Unexpected Angle Measure
				Curiosity 120. An Unexpected Equality from a Right Triangle
				Curiosity 121. The Conundrum: Perpendicular or Parallel
				Curiosity 122. Multiple Midpoints in a Right Triangle Determine Equal Line Segments
				Curiosity 123. Perpendiculars in Right Triangles that Generate Equal Angles
				Curiosity 124. Right Triangles Sharing a Common Hypotenuse Generate an Unexpected Equality
				Curiosity 125. The Pythagorean Theorem Revisited Geometrically
				Curiosity 126. Squares on Triangle Sides Produce Noteworthy Surprises
				Curiosity 127. More Properties Generated by Squares on Triangle Sides
				Curiosity 128. An Unexpected Perpendicularity
				Curiosity 129. A Surprising Concurrency from Squares on the Sides of a Triangle
				Curiosity 130. Another Surprising Concurrency from Squares on Triangle Sides
				Curiosity 131. Using Squares on Triangle Sides to Create another Surprising Concurrency
				Curiosity 132. Squares on the Legs of a Right Triangle Produce an Unexpected Equality
				Curiosity 133. More About Squares on the Legs of a Right Triangle
				Curiosity 134. More Placements of Squares on Right Triangles
				Curiosity 135. Another Unexpected Area Equality
				Curiosity 136. The Square on the Hypotenuse of a Right Triangle
				Curiosity 137. A Truly Unexpected Collinearity
				Curiosity 138. A Most Unusual Procedure to Divide a Square into Two Equal Parts
				Curiosity 139. Doubling a Square
				Curiosity 140. A Strange Construction of Parallel Lines
				Curiosity 141. An Unusual Technique to Find the Midpoint of a Line Segment
				Curiosity 142. The Unexpected Appearance of Parallel Lines
				Curiosity 143. The Unanticipated Parallel Line
				Curiosity 144. The Surprising Perpendicularity
				Curiosity 145. Another Unexpected Perpendicularity
				Curiosity 146. Yet Another Unexpected Right Angle
				Curiosity 147. Four Important Concyclic Points
				Curiosity 148. Four Remarkable Concyclic Points
				Curiosity 149. Four Unexpected Concyclic Points
				Curiosity 150. Perpendiculars that Generate Concyclic Points
				Curiosity 151. Altitudes and Circles that Generate Another Circle
				Curiosity 152. More Unexpected Concyclic Points
				Curiosity 153. A Surprising Five-Point Circle
				Curiosity 154. The Famous Nine-Point Circle
				Curiosity 155. A Collinearity with the Center of the Nine-Point Circle
				Curiosity 156. The Meeting of the Three Famous Triangle Centers
				Curiosity 157. Properties of the Nine-Point Circle
				Curiosity 158. More Properties of the Nine-Point Circle
				Curiosity 159. An Unexpected Collinearity
				Curiosity 160. A Concurrency Generated by the Orthic Triangle
				Curiosity 161. Altitudes Produce a Concurrency and Equality
				Curiosity 162. Napoleon’s Contribution to Mathematics
				Curiosity 163. Napoleon’s Minimum Distance Point (the Fermat Point)
				Curiosity 164. When the Minimum Distance Point is not Inside the Triangle
				Curiosity 165. Extensions of Napoleon’s Theorem
				Curiosity 166. Overlapping Side-Equilateral Triangles
				Curiosity 167. Surprising Triangle Area Relationship
				Curiosity 168. The Centroid Enters the Previous Configuration
				Curiosity 169. The Emergence of Another Equilateral Triangle
				Curiosity 170. A Novel Way of Finding the Center of the Circumscribed Circle
				Curiosity 171. A Concurrency Point of Circles
				Curiosity 172. The Famous Miquel Theorem
				Curiosity 173. Miquel’s Similar Triangles
				Curiosity 174. The Astounding Morley’s Theorem
				Curiosity 175. Morley’s Theorem Extended
		Proofs of the Triangle Curiosities
			Curiosity 1. Angles at the Incenter
				Curiosity 2. An Unexpected Perpendicularity
				Curiosity 3. A Most Unusual Line Bisection
				Curiosity 4. Gergonne’s Discovery Involving the Inscribed Circle of a Triangle
				Curiosity 5. Another Unexpected Concurrency for Gergonne’s Triangle
				Curiosity 6. A Novelty Concerning the Circumscribed and Inscribed Circles
				Curiosity 7. Concurrency Involving the Inscribed Circle
				Curiosity 8. A More General Concurrency Involving the Inscribed Circle
				Curiosity 9. An Inscribed Circle of a General Triangle Generates Equal Angles
				Curiosity 10. Surprising Equal Angles
				Curiosity 11. An Inscribed Circle of a Triangle Generates its Circumscribed Circle
				Curiosity 12. The Feet of the Altitudes Partition a Triangle into Three Pairs of Equal-Area Triangles
				Curiosity 13. The Noteworthy Position of the Orthocenter of a Triangle
				Curiosity 14. A Circle Intersects a Triangle to Generate Lots of Equal Segments
				Curiosity 15. A Strange Appearance of a Congruent Triangle
				Curiosity 16. The Orthocenter Joins Three Other Points in Collinearity
				Curiosity 17. The Orthocenter Appears as the Midpoint of a Line Segment
				Curiosity 18. A Conglomeration of Perpendiculars Generates Equal Line Segments
				Curiosity 19. Yet Another Unexpected Collinearity
				Curiosity 20. Introducing the Orthic Triangle
				Curiosity 21. Finding the Perimeter of the Orthic Triangle
				Curiosity 22. The Orthic Triangle is a Triangle’s Smallest Inscribed Triangle
				Curiosity 23. The Orthic Triangle’s Surprising Similar Partner
				Curiosity 24. A Nice Concurrency Generated by the Orthic Triangle
				Curiosity 25. Orthic Triangle Generates an Isosceles Triangle
				Curiosity 26. The Orthic Triangle Generates a Parallelogram
				Curiosity 27. Concyclic Points Generated by the Orthic Triangle
				Curiosity 28. The Orthic Triangle Generates Concurrent Lines
				Curiosity 29. The Orthic Triangle Generates Collinear Points
				Curiosity 30. An Unexpected Property of Altitude Feet of a Triangle
				Curiosity 31. How a Non-Isosceles Triangle Can Generate a Parallelogram
				Curiosity 32. Perpendiculars Generating Unexpected Parallel Lines
				Curiosity 33. Tangents and Inscribed-Triangle Sides Generate a Surprising Collinearity
				Curiosity 34. Creating Congruent Triangles Inscribed in the Same Circle
				Curiosity 35. An Unexpected Product Equality
				Curiosity 36. An Unusual Product of Two Sides of a Triangle
				Curiosity 37. Simson’s Theorem
				Curiosity 38. An Extension of Simson’s Theorem
				Curiosity 39. An Interesting Aspect of Simson’s Theorem
				Curiosity 40. A Parallel to the Simson Line
				Curiosity 41. Two Triangles Related by a Common Point: Circumcenter – Centroid
				Curiosity 42. Introducing the Medians of a Triangle with Some of Their Amazing Properties
				Curiosity 43. The Median of a Triangle is Equidistant from Two Vertices
				Curiosity 44. The Special Median of a Right Triangle
				Curiosity 45. Medians Partition Any Triangle into Four Congruent Triangles
				Curiosity 46. How the Centroid Helps Create a Similar Triangle
				Curiosity 47. The Centroid can Provide a Most Unusual Balance
				Curiosity 48. Distances from a Triangle’s Vertices to a Random Line
				Curiosity 49. A Special Centroid Property When Two Medians are Perpendicular
				Curiosity 50. The Centroid’s Amazing Property
				Curiosity 51. Some Median Surprises
				Curiosity 52. More Median Marvels
				Curiosity 53. Comparing Medians to Triangle Perimeters
				Curiosity 54. Median Extensions Generate Collinearity
				Curiosity 55. Two Unusual Triangles Share a Common Centroid
				Curiosity 56. The Circumscribed Circle Revisited: The Incredible Relationship of the Centers of the Circumscribed Circles of the Median Triangles
				Curiosity 57. An Astonishing Equality
				Curiosity 58. An Unexpected Simultaneous Bisection and Quadrisection in a Triangle
				Curiosity 59. Noteworthy Triangle Area Relations
				Curiosity 60. A Special Feature of a Random Point in an Equilateral Triangle
				Curiosity 61. A Special Feature of a Point Outside of an Equilateral Triangle
				Curiosity 62. Using an Equilateral Triangle to Trisect a Line Segment
				Curiosity 63. Another Way to Trisect a Line Segment with an Equilateral Triangle
				Curiosity 64. An Unusual Construction of a 30°-60°- 90° Triangle inside an Equilateral Triangle
				Curiosity 65. An Unusual Product of Segments in an Equilateral Triangle
				Curiosity 66. A Surprising Relationship between an Isosceles Triangle and an Equilateral Triangle
				Curiosity 67. An Unexpected Equality in an Isosceles Triangle
				Curiosity 68. Another Unexpected Equality in an Isosceles Triangle
				Curiosity 69. A Remarkable Geometric Equality
				Curiosity 70. Another Remarkable Geometric Equality
				Curiosity 71. An Unexpected Perpendicularity
				Curiosity 72. The Unexpected Property of an Altitude to the Side of an Isosceles Triangle
				Curiosity 73. Peculiar Property of Isosceles Triangles
				Curiosity 74. A Bizarre Connection: The Triangle Incenter on its Circumcircle
				Curiosity 75. Collinear Points Generate an Angle Bisector
				Curiosity 76. Unexpected Similar Triangles
				Curiosity 77. Perpendiculars to Four Angle Bisectors of a Triangle Reveal Four Collinear Points
				Curiosity 78. A Convoluted Bisection of a Line Segment
				Curiosity 79. Determining the Perimeter of a Triangle Without Measuring its Side Lengths
				Curiosity 80. A Strange Angle Trisection
				Curiosity 81. A Most Unexpected Equality
				Curiosity 82. A Triangle Peculiarity
				Curiosity 83. A Counterintuitive Area Equality of Triangles
				Curiosity 84. Parallel Lines Create a Double Area Triangle
				Curiosity 85. Unexpected Triangle Area Relationships
				Curiosity 86. Creating a Triangle Whose Area is Three-Quarters the Area of a Given Triangle
				Curiosity 87. An Astounding Construction: Similar Triangles Whose Area Ratio is 1:4
				Curiosity 88. Unforeseen Equality of Inscribed Triangles
				Curiosity 89. The Unanticipated Commonality of Equal Area Triangles
				Curiosity 90. How a Random Point Divides the Area of a Triangle in Half
				Curiosity 91. Determining a Triangle One-Third of the Area of a Given Triangle
				Curiosity 92. Trisection Points Partitioning a Triangle
				Curiosity 93. Further Surprises Provided by the Trisection Points of Triangle Sides
				Curiosity 94. Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 95. Yet Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 96. Medians and Trisectors Partitioning a Triangle with an Unexpected Result
				Curiosity 97. The Unforeseen Characteristic of a Random Triangle with a 60° Angle
				Curiosity 98. Another Surprising Feature of a Random Triangle with a 60° Angle
				Curiosity 99. Another Unexpected Collinearity
				Curiosity 100. Reflections of Triangles Generate Concurrent Circles and Concurrent Lines
				Curiosity 101. The Wonders of Three Concurrent Congruent Circles
				Curiosity 102. Further Unexpected Concurrencies
				Curiosity 103. One Concurrency Generates Another Concurrency
				Curiosity 104. Unusual Perpendiculars Generating Concurrencies
				Curiosity 105. A Most Unexpected Concurrency
				Curiosity 106. An Intriguing Concurrency
				Curiosity 107. Another Unexpected and Unforeseen Concurrency
				Curiosity 108. A Most Unexpected Concurrency from a Triangle
				Curiosity 109. Astounding Point Property in a Triangle
				Curiosity 110. A Counterintuitive Concurrency
				Curiosity 111. Concurrent Angle Bisectors of Triangles with a Common Base
				Curiosity 112. An Unexpected Concurrency with the Circumscribed Circle
				Curiosity 113. Another Surprising Aspect of Concurrent Cevians
				Curiosity 114. A Surprising Equality
				Curiosity 115. An Unforeseen Triangle Surprise
				Curiosity 116. A Remarkable Property of Two Triangles with a Common Base
				Curiosity 117. A Surprising Line Partitioning
				Curiosity 118. The Hidden Length Equality of Line Segments in a Triangle
				Curiosity 119. The Unexpected Angle Measure
				Curiosity 120. An Unexpected Equality from a Right Triangle
				Curiosity 121. The Conundrum: Perpendicular or Parallel
				Curiosity 122. Multiple Midpoints in a Right Triangle Determine Equal Line Segments
				Curiosity 123. Perpendiculars in Right Triangles that Generate Equal Angles
				Curiosity 124. Right Triangles Sharing a Common Hypotenuse Generate an Unexpected Equality
				Curiosity 125. The Pythagorean Theorem Revisited Geometrically
				Curiosity 126. Squares on Triangle Sides Produce Noteworthy Surprises
				Curiosity 127. More Properties Generated by Squares on Triangle Sides
				Curiosity 128. An Unexpected Perpendicularity
				Curiosity 129. A Surprising Concurrency from Squares on the Sides of a Triangle
				Curiosity 130. Another Surprising Concurrency from Squares on Triangle Sides
				Curiosity 131. Using Squares on Triangle Sides to Create another Surprising Concurrency
				Curiosity 132. Squares on the Legs of a Right Triangle Produce an Unexpected Equality
				Curiosity 133. More About Squares on the Legs of a Right Triangle
				Curiosity 134. More Placements of Squares on Right Triangles
				Curiosity 135. Another Unexpected Area Equality
				Curiosity 136. The Square on the Hypotenuse of a Right Triangle
				Curiosity 137. A Truly Unexpected Collinearity
				Curiosity 138. A Most Unusual Procedure to Divide a Square into Two Equal Parts
				Curiosity 139. Doubling a Square
				Curiosity 140. A Strange Construction of Parallel Lines
				Curiosity 141. An Unusual Technique to Find the Midpoint of a Line Segment
				Curiosity 142. The Unexpected Appearance of Parallel Lines
				Curiosity 143. The Unanticipated Parallel Line
				Curiosity 144. The Surprising Perpendicularity
				Curiosity 145. Another Unexpected Perpendicularity
				Curiosity 146. Yet Another Unexpected Right Angle
				Curiosity 147. Four Important Concyclic Points
				Curiosity 148. Four Remarkable Concyclic Points
				Curiosity 149. Four Unexpected Concyclic Points
				Curiosity 150. Perpendiculars that Generate Concyclic Points
				Curiosity 151. Altitudes and Circles that Generate Another Circle
				Curiosity 152. More Unexpected Concyclic Points
				Curiosity 153. A Surprising Five-Point Circle
				Curiosity 154. The Famous Nine-Point Circle
				Curiosity 155. A Collinearity with the Center of the Nine-Point Circle
				Curiosity 156. The Meeting of the Three Famous Triangle Centers
				Curiosity 157. Properties of the Nine-Point Circle
				Curiosity 158. More Properties of the Nine-Point Circle
				Curiosity 159. An Unexpected Collinearity
				Curiosity 160. A Concurrency Generated by the Orthic Triangle
				Curiosity 161. Altitudes Produce a Concurrency and Equality
				Curiosity 162. Napoleon’s Contribution to Mathematics
				Curiosity 163. Napoleon’s Minimum Distance Point (The Fermat Point)
				Curiosity 164. When the Minimum Distance Point is not Inside the Triangle
				Curiosity 165. Extensions of Napoleon’s Theorem
				Curiosity 166. Overlapping Side-Equilateral Triangles
				Curiosity 167. Surprising Triangle Area Relationship
				Curiosity 168. The Centroid Enters the Previous Configuration
				Curiosity 169. The Emergence of Another Equilateral Triangle
				Curiosity 170. A Novel Way of Finding the Center of the Circumscribed Circle
				Curiosity 171. A Concurrency Point of Circles
				Curiosity 172. The Famous Miquel Theorem
				Curiosity 173. Miquel’s Similar Triangles
				Curiosity 174. The Astounding Morley’s Theorem
				Curiosity 175. Morley’s Theorem Extended
		Toolbox
			Introduction: The Geometry Toolbox
			A.	Tools You Are Probably Familiar with from the High School Geometry Course
				A1: Congruence of Triangles
				A2: Similarity of Triangles
				A3: Right Triangle Properties (See Figure A3)
				A4: Angles Related to a Circle (see Figure A4)
				A5:	Tangents, Secants, and Chords: Segments of a Circle (see Figure A5)
				A6: The Law of Sines and the Law of Cosines
				A7: Angle Sum and Difference Identities
			B.	Less Familiar Tools—However, Useful and Fascinating
				B1. Interior Angle Bisector in a Triangle
				B2. Exterior Angle Bisector in a Triangle
				B3. Menelaus’ Theorem
				B4. Ceva’s Theorem
				B5. The Trigonometric Version of Ceva’s Theorem
				B6. Ceva’s Theorem Extended
				B7. Desargues’s Theorem
				B8. Stewart’s Theorem
				B9. Theorem of Apollonius
				B10. Ptolemy’s Theorem
				B11. Isometries: Reflection and Rotation
				B12. Homothety and Similarity
				B13. Polarity on Circles
	Index
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	Contents
	Acknowledgments
	About the Authors
		Introduction
		Geometric Curiosities: Introducing the Triangle
			Curiosity 1. Angles at the Incenter
				Curiosity 2. An Unexpected Perpendicularity
				Curiosity 3. A Most Unusual Line Bisection
				Curiosity 4. Gergonne’s Discovery Involving the Inscribed Circle of a Triangle
				Curiosity 5. Another Unexpected Concurrency for Gergonne’s Triangle
				Curiosity 6. A Novelty Concerning the Circumscribed and Inscribed Circles
				Curiosity 7. Concurrency Involving the Inscribed Circle
				Curiosity 8. A More General Concurrency Involving the Inscribed Circle
				Curiosity 9. An Inscribed Circle of a General Triangle Generates Equal Angles
				Curiosity 10. Surprising Equal Angles
				Curiosity 11. An Inscribed Circle of a Triangle Generates its Circumscribed Circle
				Curiosity 12. The Feet of the Altitudes Partition a Triangle into Three Pairs of Equal-Area Triangles
				Curiosity 13. The Noteworthy Position of the Orthocenter of a Triangle
				Curiosity 14. A Circle Intersects a Triangle to Generate Lots of Equal Segments
				Curiosity 15. A Strange Appearance of a Congruent Triangle
				Curiosity 16. The Orthocenter Joins Three Other Points in Collinearity
				Curiosity 17. The Orthocenter Appears as the Midpoint of a Line Segment
				Curiosity 18. A Conglomeration of Perpendiculars Generates Equal Line Segments
				Curiosity 19. Yet Another Unexpected Collinearity
				Curiosity 20. Introducing the Orthic Triangle
				Curiosity 21. Finding the Perimeter of the Orthic Triangle
				Curiosity 22. The Orthic Triangle is a Triangle’s Smallest Inscribed Triangle
				Curiosity 23. The Orthic Triangle’s Surprising Similar Partner
				Curiosity 24. A Nice Concurrency Generated by the Orthic Triangle
				Curiosity 25. Orthic Triangle Generates an Isosceles Triangle
				Curiosity 26. The Orthic Triangle Generates a Parallelogram
				Curiosity 27. Concyclic Points Generated by the Orthic Triangle
				Curiosity 28. The Orthic Triangle Generates Concurrent Lines
				Curiosity 29. The Orthic Triangle Generates Collinear Points
				Curiosity 30. An Unexpected Property of Altitude Feet of a Triangle
				Curiosity 31. How a Non-Isosceles Triangle Can Generate a Parallelogram
				Curiosity 32. Perpendiculars Generating Unexpected Parallel Lines
				Curiosity 33. Inscribed Triangle Generates Collinearity
				Curiosity 34. Unexpected Congruent Triangles in a Circle
				Curiosity 35. An Unexpected Product Equality
				Curiosity 36. An Unusual Product of Two Sides of a Triangle
				Curiosity 37. Simson’s Theorem
				Curiosity 38. An Extension of Simson’s Theorem
				Curiosity 39. An Interesting Aspect of Simson’s Theorem
				Curiosity 40. A Parallel to the Simson Line
				Curiosity 41. Two Triangles Related by a Common Point: Circumcenter – Centroid
				Curiosity 42. Introducing the Medians of a Triangle with Some of Their Amazing Properties
				Curiosity 43. The Median of a Triangle is Equidistant from Two Vertices
				Curiosity 44. The Special Median of a Right Triangle
				Curiosity 45. Medians Partition Any Triangle into Four Congruent Triangles
				Curiosity 46. How the Centroid Helps Create a Similar Triangle
				Curiosity 47. The Centroid can Provide a Most Unusual Balance
				Curiosity 48. Distances from a Triangle’s Vertices to a Random Line
				Curiosity 49. A Special Centroid Property When Two Medians are Perpendicular
				Curiosity 50. The Centroid’s Amazing Property
				Curiosity 51. Some Median Surprises
				Curiosity 52. More Median Marvels
				Curiosity 53. Comparing Medians to Triangle Perimeters
				Curiosity 54. Median Extensions Generate Collinearity
				Curiosity 55. Two Unusual Triangles Share a Common Centroid
				Curiosity 56. The Circumscribed Circle Revisited: The Incredible Relationship of the Centers of the Circumscribed Circles of the Median Triangles
				Curiosity 57. An Astonishing Equality
				Curiosity 58. An Unexpected Simultaneous Bisection and Quadrisection in a Triangle
				Curiosity 59. Noteworthy Triangle Area Relations
				Curiosity 60. A Special Feature of a Random Point in an Equilateral Triangle
				Curiosity 61. A Special Feature of a Point Outside of an Equilateral Triangle
				Curiosity 62. Using an Equilateral Triangle to Trisect a Line Segment
				Curiosity 63. Another Way to Trisect a Line Segment with an Equilateral Triangle
				Curiosity 64. An Unusual Construction of a 30°-60°-90° Triangle Inside an Equilateral Triangle
				Curiosity 65. An Unusual Product of Segments in an Equilateral Triangle
				Curiosity 66. A Surprising Relationship Between an Isosceles Triangle and an Equilateral Triangle
				Curiosity 67. An Unexpected Equality in an Isosceles Triangle
				Curiosity 68. Another Unexpected Equality in an Isosceles Triangle
				Curiosity 69. A Remarkable Geometric Equality
				Curiosity 70. Another Remarkable Geometric Equality
				Curiosity 71. An Unexpected Perpendicularity
				Curiosity 72. The Unexpected Property of an Altitude to the Side of an Isosceles Triangle
				Curiosity 73. Peculiar Property of Isosceles Triangles
				Curiosity 74. A Bizarre Connection: The Triangle Incenter on its Circumcircle
				Curiosity 75. Collinear Points Generate an Angle Bisector
				Curiosity 76. Unexpected Similar Triangles
				Curiosity 77. Perpendiculars to Four Angle Bisectors of a Triangle Reveal Four Collinear Points
				Curiosity 78. A Convoluted Bisection of a Line Segment
				Curiosity 79. Determining the Perimeter of a Triangle Without Measuring its Side Lengths
				Curiosity 80. A Strange Angle Trisection
				Curiosity 81. A Most Unexpected Equality
				Curiosity 82. A Triangle Peculiarity
				Curiosity 83. A Counterintuitive Area Equality of Triangles
				Curiosity 84. Parallel Lines Create a Double Area Triangle
				Curiosity 85. Unexpected Triangle Area Relationships
				Curiosity 86. Creating a Triangle Whose Area is Three-Quarters the Area of a Given Triangle
				Curiosity 87. An Astounding Construction: Similar Triangles Whose Area Ratio is 1:4
				Curiosity 88. Unforeseen Equality of Inscribed Triangles
				Curiosity 89. The Unanticipated Commonality of Equal Area Triangles
				Curiosity 90. How a Random Point Divides the Area of a Triangle in Half
				Curiosity 91. Determining a Triangle One-Third of the Area of a Given Triangle
				Curiosity 92. Trisection Points Partitioning a Triangle
				Curiosity 93. Further Surprises Provided by the Trisection Points of Triangle Sides
				Curiosity 94. Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 95. Yet Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 96. Medians and Trisectors Partitioning a Triangle with an Unexpected Result
				Curiosity 97. The Unforeseen Characteristic of a Random Triangle with a 60° Angle
				Curiosity 98. Another Surprising Feature of a Random Triangle with a 60° Angle
				Curiosity 99. Another Unexpected Collinearity
				Curiosity 100. Reflections of Triangles Generate Concurrent Circles and Concurrent Lines
				Curiosity 101. The Wonders of Three Concurrent Congruent Circles
				Curiosity 102. Further Unexpected Concurrencies
				Curiosity 103. One Concurrency Generates Another Concurrency
				Curiosity 104. Unusual Perpendiculars Generating Concurrencies
				Curiosity 105. A Most Unexpected Concurrency
				Curiosity 106. An Intriguing Concurrency
				Curiosity 107. Another Unexpected and Unforeseen Concurrency
				Curiosity 108. A Most Unexpected Concurrency from a Triangle
				Curiosity 109. Astounding Point Property in a Triangle
				Curiosity 110. A Counterintuitive Concurrency
				Curiosity 111. Concurrent Angle Bisectors of Triangles with a Common Base
				Curiosity 112. An Unexpected Concurrency with the Circumscribed Circle
				Curiosity 113. Another Surprising Aspect of Concurrent Cevians
				Curiosity 114. A Surprising Equality
				Curiosity 115. An Unforeseen Triangle Surprise
				Curiosity 116. A Remarkable Property of Two Triangles with a Common Base
				Curiosity 117. A Surprising Line Partitioning
				Curiosity 118. The Hidden Length Equality of Line Segments in a Triangle
				Curiosity 119. The Unexpected Angle Measure
				Curiosity 120. An Unexpected Equality from a Right Triangle
				Curiosity 121. The Conundrum: Perpendicular or Parallel
				Curiosity 122. Multiple Midpoints in a Right Triangle Determine Equal Line Segments
				Curiosity 123. Perpendiculars in Right Triangles that Generate Equal Angles
				Curiosity 124. Right Triangles Sharing a Common Hypotenuse Generate an Unexpected Equality
				Curiosity 125. The Pythagorean Theorem Revisited Geometrically
				Curiosity 126. Squares on Triangle Sides Produce Noteworthy Surprises
				Curiosity 127. More Properties Generated by Squares on Triangle Sides
				Curiosity 128. An Unexpected Perpendicularity
				Curiosity 129. A Surprising Concurrency from Squares on the Sides of a Triangle
				Curiosity 130. Another Surprising Concurrency from Squares on Triangle Sides
				Curiosity 131. Using Squares on Triangle Sides to Create another Surprising Concurrency
				Curiosity 132. Squares on the Legs of a Right Triangle Produce an Unexpected Equality
				Curiosity 133. More About Squares on the Legs of a Right Triangle
				Curiosity 134. More Placements of Squares on Right Triangles
				Curiosity 135. Another Unexpected Area Equality
				Curiosity 136. The Square on the Hypotenuse of a Right Triangle
				Curiosity 137. A Truly Unexpected Collinearity
				Curiosity 138. A Most Unusual Procedure to Divide a Square into Two Equal Parts
				Curiosity 139. Doubling a Square
				Curiosity 140. A Strange Construction of Parallel Lines
				Curiosity 141. An Unusual Technique to Find the Midpoint of a Line Segment
				Curiosity 142. The Unexpected Appearance of Parallel Lines
				Curiosity 143. The Unanticipated Parallel Line
				Curiosity 144. The Surprising Perpendicularity
				Curiosity 145. Another Unexpected Perpendicularity
				Curiosity 146. Yet Another Unexpected Right Angle
				Curiosity 147. Four Important Concyclic Points
				Curiosity 148. Four Remarkable Concyclic Points
				Curiosity 149. Four Unexpected Concyclic Points
				Curiosity 150. Perpendiculars that Generate Concyclic Points
				Curiosity 151. Altitudes and Circles that Generate Another Circle
				Curiosity 152. More Unexpected Concyclic Points
				Curiosity 153. A Surprising Five-Point Circle
				Curiosity 154. The Famous Nine-Point Circle
				Curiosity 155. A Collinearity with the Center of the Nine-Point Circle
				Curiosity 156. The Meeting of the Three Famous Triangle Centers
				Curiosity 157. Properties of the Nine-Point Circle
				Curiosity 158. More Properties of the Nine-Point Circle
				Curiosity 159. An Unexpected Collinearity
				Curiosity 160. A Concurrency Generated by the Orthic Triangle
				Curiosity 161. Altitudes Produce a Concurrency and Equality
				Curiosity 162. Napoleon’s Contribution to Mathematics
				Curiosity 163. Napoleon’s Minimum Distance Point (the Fermat Point)
				Curiosity 164. When the Minimum Distance Point is not Inside the Triangle
				Curiosity 165. Extensions of Napoleon’s Theorem
				Curiosity 166. Overlapping Side-Equilateral Triangles
				Curiosity 167. Surprising Triangle Area Relationship
				Curiosity 168. The Centroid Enters the Previous Configuration
				Curiosity 169. The Emergence of Another Equilateral Triangle
				Curiosity 170. A Novel Way of Finding the Center of the Circumscribed Circle
				Curiosity 171. A Concurrency Point of Circles
				Curiosity 172. The Famous Miquel Theorem
				Curiosity 173. Miquel’s Similar Triangles
				Curiosity 174. The Astounding Morley’s Theorem
				Curiosity 175. Morley’s Theorem Extended
		Proofs of the Triangle Curiosities
			Curiosity 1. Angles at the Incenter
				Curiosity 2. An Unexpected Perpendicularity
				Curiosity 3. A Most Unusual Line Bisection
				Curiosity 4. Gergonne’s Discovery Involving the Inscribed Circle of a Triangle
				Curiosity 5. Another Unexpected Concurrency for Gergonne’s Triangle
				Curiosity 6. A Novelty Concerning the Circumscribed and Inscribed Circles
				Curiosity 7. Concurrency Involving the Inscribed Circle
				Curiosity 8. A More General Concurrency Involving the Inscribed Circle
				Curiosity 9. An Inscribed Circle of a General Triangle Generates Equal Angles
				Curiosity 10. Surprising Equal Angles
				Curiosity 11. An Inscribed Circle of a Triangle Generates its Circumscribed Circle
				Curiosity 12. The Feet of the Altitudes Partition a Triangle into Three Pairs of Equal-Area Triangles
				Curiosity 13. The Noteworthy Position of the Orthocenter of a Triangle
				Curiosity 14. A Circle Intersects a Triangle to Generate Lots of Equal Segments
				Curiosity 15. A Strange Appearance of a Congruent Triangle
				Curiosity 16. The Orthocenter Joins Three Other Points in Collinearity
				Curiosity 17. The Orthocenter Appears as the Midpoint of a Line Segment
				Curiosity 18. A Conglomeration of Perpendiculars Generates Equal Line Segments
				Curiosity 19. Yet Another Unexpected Collinearity
				Curiosity 20. Introducing the Orthic Triangle
				Curiosity 21. Finding the Perimeter of the Orthic Triangle
				Curiosity 22. The Orthic Triangle is a Triangle’s Smallest Inscribed Triangle
				Curiosity 23. The Orthic Triangle’s Surprising Similar Partner
				Curiosity 24. A Nice Concurrency Generated by the Orthic Triangle
				Curiosity 25. Orthic Triangle Generates an Isosceles Triangle
				Curiosity 26. The Orthic Triangle Generates a Parallelogram
				Curiosity 27. Concyclic Points Generated by the Orthic Triangle
				Curiosity 28. The Orthic Triangle Generates Concurrent Lines
				Curiosity 29. The Orthic Triangle Generates Collinear Points
				Curiosity 30. An Unexpected Property of Altitude Feet of a Triangle
				Curiosity 31. How a Non-Isosceles Triangle Can Generate a Parallelogram
				Curiosity 32. Perpendiculars Generating Unexpected Parallel Lines
				Curiosity 33. Tangents and Inscribed-Triangle Sides Generate a Surprising Collinearity
				Curiosity 34. Creating Congruent Triangles Inscribed in the Same Circle
				Curiosity 35. An Unexpected Product Equality
				Curiosity 36. An Unusual Product of Two Sides of a Triangle
				Curiosity 37. Simson’s Theorem
				Curiosity 38. An Extension of Simson’s Theorem
				Curiosity 39. An Interesting Aspect of Simson’s Theorem
				Curiosity 40. A Parallel to the Simson Line
				Curiosity 41. Two Triangles Related by a Common Point: Circumcenter – Centroid
				Curiosity 42. Introducing the Medians of a Triangle with Some of Their Amazing Properties
				Curiosity 43. The Median of a Triangle is Equidistant from Two Vertices
				Curiosity 44. The Special Median of a Right Triangle
				Curiosity 45. Medians Partition Any Triangle into Four Congruent Triangles
				Curiosity 46. How the Centroid Helps Create a Similar Triangle
				Curiosity 47. The Centroid can Provide a Most Unusual Balance
				Curiosity 48. Distances from a Triangle’s Vertices to a Random Line
				Curiosity 49. A Special Centroid Property When Two Medians are Perpendicular
				Curiosity 50. The Centroid’s Amazing Property
				Curiosity 51. Some Median Surprises
				Curiosity 52. More Median Marvels
				Curiosity 53. Comparing Medians to Triangle Perimeters
				Curiosity 54. Median Extensions Generate Collinearity
				Curiosity 55. Two Unusual Triangles Share a Common Centroid
				Curiosity 56. The Circumscribed Circle Revisited: The Incredible Relationship of the Centers of the Circumscribed Circles of the Median Triangles
				Curiosity 57. An Astonishing Equality
				Curiosity 58. An Unexpected Simultaneous Bisection and Quadrisection in a Triangle
				Curiosity 59. Noteworthy Triangle Area Relations
				Curiosity 60. A Special Feature of a Random Point in an Equilateral Triangle
				Curiosity 61. A Special Feature of a Point Outside of an Equilateral Triangle
				Curiosity 62. Using an Equilateral Triangle to Trisect a Line Segment
				Curiosity 63. Another Way to Trisect a Line Segment with an Equilateral Triangle
				Curiosity 64. An Unusual Construction of a 30°-60°- 90° Triangle inside an Equilateral Triangle
				Curiosity 65. An Unusual Product of Segments in an Equilateral Triangle
				Curiosity 66. A Surprising Relationship between an Isosceles Triangle and an Equilateral Triangle
				Curiosity 67. An Unexpected Equality in an Isosceles Triangle
				Curiosity 68. Another Unexpected Equality in an Isosceles Triangle
				Curiosity 69. A Remarkable Geometric Equality
				Curiosity 70. Another Remarkable Geometric Equality
				Curiosity 71. An Unexpected Perpendicularity
				Curiosity 72. The Unexpected Property of an Altitude to the Side of an Isosceles Triangle
				Curiosity 73. Peculiar Property of Isosceles Triangles
				Curiosity 74. A Bizarre Connection: The Triangle Incenter on its Circumcircle
				Curiosity 75. Collinear Points Generate an Angle Bisector
				Curiosity 76. Unexpected Similar Triangles
				Curiosity 77. Perpendiculars to Four Angle Bisectors of a Triangle Reveal Four Collinear Points
				Curiosity 78. A Convoluted Bisection of a Line Segment
				Curiosity 79. Determining the Perimeter of a Triangle Without Measuring its Side Lengths
				Curiosity 80. A Strange Angle Trisection
				Curiosity 81. A Most Unexpected Equality
				Curiosity 82. A Triangle Peculiarity
				Curiosity 83. A Counterintuitive Area Equality of Triangles
				Curiosity 84. Parallel Lines Create a Double Area Triangle
				Curiosity 85. Unexpected Triangle Area Relationships
				Curiosity 86. Creating a Triangle Whose Area is Three-Quarters the Area of a Given Triangle
				Curiosity 87. An Astounding Construction: Similar Triangles Whose Area Ratio is 1:4
				Curiosity 88. Unforeseen Equality of Inscribed Triangles
				Curiosity 89. The Unanticipated Commonality of Equal Area Triangles
				Curiosity 90. How a Random Point Divides the Area of a Triangle in Half
				Curiosity 91. Determining a Triangle One-Third of the Area of a Given Triangle
				Curiosity 92. Trisection Points Partitioning a Triangle
				Curiosity 93. Further Surprises Provided by the Trisection Points of Triangle Sides
				Curiosity 94. Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 95. Yet Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 96. Medians and Trisectors Partitioning a Triangle with an Unexpected Result
				Curiosity 97. The Unforeseen Characteristic of a Random Triangle with a 60° Angle
				Curiosity 98. Another Surprising Feature of a Random Triangle with a 60° Angle
				Curiosity 99. Another Unexpected Collinearity
				Curiosity 100. Reflections of Triangles Generate Concurrent Circles and Concurrent Lines
				Curiosity 101. The Wonders of Three Concurrent Congruent Circles
				Curiosity 102. Further Unexpected Concurrencies
				Curiosity 103. One Concurrency Generates Another Concurrency
				Curiosity 104. Unusual Perpendiculars Generating Concurrencies
				Curiosity 105. A Most Unexpected Concurrency
				Curiosity 106. An Intriguing Concurrency
				Curiosity 107. Another Unexpected and Unforeseen Concurrency
				Curiosity 108. A Most Unexpected Concurrency from a Triangle
				Curiosity 109. Astounding Point Property in a Triangle
				Curiosity 110. A Counterintuitive Concurrency
				Curiosity 111. Concurrent Angle Bisectors of Triangles with a Common Base
				Curiosity 112. An Unexpected Concurrency with the Circumscribed Circle
				Curiosity 113. Another Surprising Aspect of Concurrent Cevians
				Curiosity 114. A Surprising Equality
				Curiosity 115. An Unforeseen Triangle Surprise
				Curiosity 116. A Remarkable Property of Two Triangles with a Common Base
				Curiosity 117. A Surprising Line Partitioning
				Curiosity 118. The Hidden Length Equality of Line Segments in a Triangle
				Curiosity 119. The Unexpected Angle Measure
				Curiosity 120. An Unexpected Equality from a Right Triangle
				Curiosity 121. The Conundrum: Perpendicular or Parallel
				Curiosity 122. Multiple Midpoints in a Right Triangle Determine Equal Line Segments
				Curiosity 123. Perpendiculars in Right Triangles that Generate Equal Angles
				Curiosity 124. Right Triangles Sharing a Common Hypotenuse Generate an Unexpected Equality
				Curiosity 125. The Pythagorean Theorem Revisited Geometrically
				Curiosity 126. Squares on Triangle Sides Produce Noteworthy Surprises
				Curiosity 127. More Properties Generated by Squares on Triangle Sides
				Curiosity 128. An Unexpected Perpendicularity
				Curiosity 129. A Surprising Concurrency from Squares on the Sides of a Triangle
				Curiosity 130. Another Surprising Concurrency from Squares on Triangle Sides
				Curiosity 131. Using Squares on Triangle Sides to Create another Surprising Concurrency
				Curiosity 132. Squares on the Legs of a Right Triangle Produce an Unexpected Equality
				Curiosity 133. More About Squares on the Legs of a Right Triangle
				Curiosity 134. More Placements of Squares on Right Triangles
				Curiosity 135. Another Unexpected Area Equality
				Curiosity 136. The Square on the Hypotenuse of a Right Triangle
				Curiosity 137. A Truly Unexpected Collinearity
				Curiosity 138. A Most Unusual Procedure to Divide a Square into Two Equal Parts
				Curiosity 139. Doubling a Square
				Curiosity 140. A Strange Construction of Parallel Lines
				Curiosity 141. An Unusual Technique to Find the Midpoint of a Line Segment
				Curiosity 142. The Unexpected Appearance of Parallel Lines
				Curiosity 143. The Unanticipated Parallel Line
				Curiosity 144. The Surprising Perpendicularity
				Curiosity 145. Another Unexpected Perpendicularity
				Curiosity 146. Yet Another Unexpected Right Angle
				Curiosity 147. Four Important Concyclic Points
				Curiosity 148. Four Remarkable Concyclic Points
				Curiosity 149. Four Unexpected Concyclic Points
				Curiosity 150. Perpendiculars that Generate Concyclic Points
				Curiosity 151. Altitudes and Circles that Generate Another Circle
				Curiosity 152. More Unexpected Concyclic Points
				Curiosity 153. A Surprising Five-Point Circle
				Curiosity 154. The Famous Nine-Point Circle
				Curiosity 155. A Collinearity with the Center of the Nine-Point Circle
				Curiosity 156. The Meeting of the Three Famous Triangle Centers
				Curiosity 157. Properties of the Nine-Point Circle
				Curiosity 158. More Properties of the Nine-Point Circle
				Curiosity 159. An Unexpected Collinearity
				Curiosity 160. A Concurrency Generated by the Orthic Triangle
				Curiosity 161. Altitudes Produce a Concurrency and Equality
				Curiosity 162. Napoleon’s Contribution to Mathematics
				Curiosity 163. Napoleon’s Minimum Distance Point (The Fermat Point)
				Curiosity 164. When the Minimum Distance Point is not Inside the Triangle
				Curiosity 165. Extensions of Napoleon’s Theorem
				Curiosity 166. Overlapping Side-Equilateral Triangles
				Curiosity 167. Surprising Triangle Area Relationship
				Curiosity 168. The Centroid Enters the Previous Configuration
				Curiosity 169. The Emergence of Another Equilateral Triangle
				Curiosity 170. A Novel Way of Finding the Center of the Circumscribed Circle
				Curiosity 171. A Concurrency Point of Circles
				Curiosity 172. The Famous Miquel Theorem
				Curiosity 173. Miquel’s Similar Triangles
				Curiosity 174. The Astounding Morley’s Theorem
				Curiosity 175. Morley’s Theorem Extended
		Toolbox
			Introduction: The Geometry Toolbox
			A.	Tools You Are Probably Familiar with from the High School Geometry Course
				A1: Congruence of Triangles
				A2: Similarity of Triangles
				A3: Right Triangle Properties (See Figure A3)
				A4: Angles Related to a Circle (see Figure A4)
				A5:	Tangents, Secants, and Chords: Segments of a Circle (see Figure A5)
				A6: The Law of Sines and the Law of Cosines
				A7: Angle Sum and Difference Identities
			B.	Less Familiar Tools—However, Useful and Fascinating
				B1. Interior Angle Bisector in a Triangle
				B2. Exterior Angle Bisector in a Triangle
				B3. Menelaus’ Theorem
				B4. Ceva’s Theorem
				B5. The Trigonometric Version of Ceva’s Theorem
				B6. Ceva’s Theorem Extended
				B7. Desargues’s Theorem
				B8. Stewart’s Theorem
				B9. Theorem of Apollonius
				B10. Ptolemy’s Theorem
				B11. Isometries: Reflection and Rotation
				B12. Homothety and Similarity
				B13. Polarity on Circles
	Index
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	Contents
	Acknowledgments
	About the Authors
		Introduction
		Geometric Curiosities: Introducing the Triangle
			Curiosity 1. Angles at the Incenter
				Curiosity 2. An Unexpected Perpendicularity
				Curiosity 3. A Most Unusual Line Bisection
				Curiosity 4. Gergonne’s Discovery Involving the Inscribed Circle of a Triangle
				Curiosity 5. Another Unexpected Concurrency for Gergonne’s Triangle
				Curiosity 6. A Novelty Concerning the Circumscribed and Inscribed Circles
				Curiosity 7. Concurrency Involving the Inscribed Circle
				Curiosity 8. A More General Concurrency Involving the Inscribed Circle
				Curiosity 9. An Inscribed Circle of a General Triangle Generates Equal Angles
				Curiosity 10. Surprising Equal Angles
				Curiosity 11. An Inscribed Circle of a Triangle Generates its Circumscribed Circle
				Curiosity 12. The Feet of the Altitudes Partition a Triangle into Three Pairs of Equal-Area Triangles
				Curiosity 13. The Noteworthy Position of the Orthocenter of a Triangle
				Curiosity 14. A Circle Intersects a Triangle to Generate Lots of Equal Segments
				Curiosity 15. A Strange Appearance of a Congruent Triangle
				Curiosity 16. The Orthocenter Joins Three Other Points in Collinearity
				Curiosity 17. The Orthocenter Appears as the Midpoint of a Line Segment
				Curiosity 18. A Conglomeration of Perpendiculars Generates Equal Line Segments
				Curiosity 19. Yet Another Unexpected Collinearity
				Curiosity 20. Introducing the Orthic Triangle
				Curiosity 21. Finding the Perimeter of the Orthic Triangle
				Curiosity 22. The Orthic Triangle is a Triangle’s Smallest Inscribed Triangle
				Curiosity 23. The Orthic Triangle’s Surprising Similar Partner
				Curiosity 24. A Nice Concurrency Generated by the Orthic Triangle
				Curiosity 25. Orthic Triangle Generates an Isosceles Triangle
				Curiosity 26. The Orthic Triangle Generates a Parallelogram
				Curiosity 27. Concyclic Points Generated by the Orthic Triangle
				Curiosity 28. The Orthic Triangle Generates Concurrent Lines
				Curiosity 29. The Orthic Triangle Generates Collinear Points
				Curiosity 30. An Unexpected Property of Altitude Feet of a Triangle
				Curiosity 31. How a Non-Isosceles Triangle Can Generate a Parallelogram
				Curiosity 32. Perpendiculars Generating Unexpected Parallel Lines
				Curiosity 33. Inscribed Triangle Generates Collinearity
				Curiosity 34. Unexpected Congruent Triangles in a Circle
				Curiosity 35. An Unexpected Product Equality
				Curiosity 36. An Unusual Product of Two Sides of a Triangle
				Curiosity 37. Simson’s Theorem
				Curiosity 38. An Extension of Simson’s Theorem
				Curiosity 39. An Interesting Aspect of Simson’s Theorem
				Curiosity 40. A Parallel to the Simson Line
				Curiosity 41. Two Triangles Related by a Common Point: Circumcenter – Centroid
				Curiosity 42. Introducing the Medians of a Triangle with Some of Their Amazing Properties
				Curiosity 43. The Median of a Triangle is Equidistant from Two Vertices
				Curiosity 44. The Special Median of a Right Triangle
				Curiosity 45. Medians Partition Any Triangle into Four Congruent Triangles
				Curiosity 46. How the Centroid Helps Create a Similar Triangle
				Curiosity 47. The Centroid can Provide a Most Unusual Balance
				Curiosity 48. Distances from a Triangle’s Vertices to a Random Line
				Curiosity 49. A Special Centroid Property When Two Medians are Perpendicular
				Curiosity 50. The Centroid’s Amazing Property
				Curiosity 51. Some Median Surprises
				Curiosity 52. More Median Marvels
				Curiosity 53. Comparing Medians to Triangle Perimeters
				Curiosity 54. Median Extensions Generate Collinearity
				Curiosity 55. Two Unusual Triangles Share a Common Centroid
				Curiosity 56. The Circumscribed Circle Revisited: The Incredible Relationship of the Centers of the Circumscribed Circles of the Median Triangles
				Curiosity 57. An Astonishing Equality
				Curiosity 58. An Unexpected Simultaneous Bisection and Quadrisection in a Triangle
				Curiosity 59. Noteworthy Triangle Area Relations
				Curiosity 60. A Special Feature of a Random Point in an Equilateral Triangle
				Curiosity 61. A Special Feature of a Point Outside of an Equilateral Triangle
				Curiosity 62. Using an Equilateral Triangle to Trisect a Line Segment
				Curiosity 63. Another Way to Trisect a Line Segment with an Equilateral Triangle
				Curiosity 64. An Unusual Construction of a 30°-60°-90° Triangle Inside an Equilateral Triangle
				Curiosity 65. An Unusual Product of Segments in an Equilateral Triangle
				Curiosity 66. A Surprising Relationship Between an Isosceles Triangle and an Equilateral Triangle
				Curiosity 67. An Unexpected Equality in an Isosceles Triangle
				Curiosity 68. Another Unexpected Equality in an Isosceles Triangle
				Curiosity 69. A Remarkable Geometric Equality
				Curiosity 70. Another Remarkable Geometric Equality
				Curiosity 71. An Unexpected Perpendicularity
				Curiosity 72. The Unexpected Property of an Altitude to the Side of an Isosceles Triangle
				Curiosity 73. Peculiar Property of Isosceles Triangles
				Curiosity 74. A Bizarre Connection: The Triangle Incenter on its Circumcircle
				Curiosity 75. Collinear Points Generate an Angle Bisector
				Curiosity 76. Unexpected Similar Triangles
				Curiosity 77. Perpendiculars to Four Angle Bisectors of a Triangle Reveal Four Collinear Points
				Curiosity 78. A Convoluted Bisection of a Line Segment
				Curiosity 79. Determining the Perimeter of a Triangle Without Measuring its Side Lengths
				Curiosity 80. A Strange Angle Trisection
				Curiosity 81. A Most Unexpected Equality
				Curiosity 82. A Triangle Peculiarity
				Curiosity 83. A Counterintuitive Area Equality of Triangles
				Curiosity 84. Parallel Lines Create a Double Area Triangle
				Curiosity 85. Unexpected Triangle Area Relationships
				Curiosity 86. Creating a Triangle Whose Area is Three-Quarters the Area of a Given Triangle
				Curiosity 87. An Astounding Construction: Similar Triangles Whose Area Ratio is 1:4
				Curiosity 88. Unforeseen Equality of Inscribed Triangles
				Curiosity 89. The Unanticipated Commonality of Equal Area Triangles
				Curiosity 90. How a Random Point Divides the Area of a Triangle in Half
				Curiosity 91. Determining a Triangle One-Third of the Area of a Given Triangle
				Curiosity 92. Trisection Points Partitioning a Triangle
				Curiosity 93. Further Surprises Provided by the Trisection Points of Triangle Sides
				Curiosity 94. Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 95. Yet Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 96. Medians and Trisectors Partitioning a Triangle with an Unexpected Result
				Curiosity 97. The Unforeseen Characteristic of a Random Triangle with a 60° Angle
				Curiosity 98. Another Surprising Feature of a Random Triangle with a 60° Angle
				Curiosity 99. Another Unexpected Collinearity
				Curiosity 100. Reflections of Triangles Generate Concurrent Circles and Concurrent Lines
				Curiosity 101. The Wonders of Three Concurrent Congruent Circles
				Curiosity 102. Further Unexpected Concurrencies
				Curiosity 103. One Concurrency Generates Another Concurrency
				Curiosity 104. Unusual Perpendiculars Generating Concurrencies
				Curiosity 105. A Most Unexpected Concurrency
				Curiosity 106. An Intriguing Concurrency
				Curiosity 107. Another Unexpected and Unforeseen Concurrency
				Curiosity 108. A Most Unexpected Concurrency from a Triangle
				Curiosity 109. Astounding Point Property in a Triangle
				Curiosity 110. A Counterintuitive Concurrency
				Curiosity 111. Concurrent Angle Bisectors of Triangles with a Common Base
				Curiosity 112. An Unexpected Concurrency with the Circumscribed Circle
				Curiosity 113. Another Surprising Aspect of Concurrent Cevians
				Curiosity 114. A Surprising Equality
				Curiosity 115. An Unforeseen Triangle Surprise
				Curiosity 116. A Remarkable Property of Two Triangles with a Common Base
				Curiosity 117. A Surprising Line Partitioning
				Curiosity 118. The Hidden Length Equality of Line Segments in a Triangle
				Curiosity 119. The Unexpected Angle Measure
				Curiosity 120. An Unexpected Equality from a Right Triangle
				Curiosity 121. The Conundrum: Perpendicular or Parallel
				Curiosity 122. Multiple Midpoints in a Right Triangle Determine Equal Line Segments
				Curiosity 123. Perpendiculars in Right Triangles that Generate Equal Angles
				Curiosity 124. Right Triangles Sharing a Common Hypotenuse Generate an Unexpected Equality
				Curiosity 125. The Pythagorean Theorem Revisited Geometrically
				Curiosity 126. Squares on Triangle Sides Produce Noteworthy Surprises
				Curiosity 127. More Properties Generated by Squares on Triangle Sides
				Curiosity 128. An Unexpected Perpendicularity
				Curiosity 129. A Surprising Concurrency from Squares on the Sides of a Triangle
				Curiosity 130. Another Surprising Concurrency from Squares on Triangle Sides
				Curiosity 131. Using Squares on Triangle Sides to Create another Surprising Concurrency
				Curiosity 132. Squares on the Legs of a Right Triangle Produce an Unexpected Equality
				Curiosity 133. More About Squares on the Legs of a Right Triangle
				Curiosity 134. More Placements of Squares on Right Triangles
				Curiosity 135. Another Unexpected Area Equality
				Curiosity 136. The Square on the Hypotenuse of a Right Triangle
				Curiosity 137. A Truly Unexpected Collinearity
				Curiosity 138. A Most Unusual Procedure to Divide a Square into Two Equal Parts
				Curiosity 139. Doubling a Square
				Curiosity 140. A Strange Construction of Parallel Lines
				Curiosity 141. An Unusual Technique to Find the Midpoint of a Line Segment
				Curiosity 142. The Unexpected Appearance of Parallel Lines
				Curiosity 143. The Unanticipated Parallel Line
				Curiosity 144. The Surprising Perpendicularity
				Curiosity 145. Another Unexpected Perpendicularity
				Curiosity 146. Yet Another Unexpected Right Angle
				Curiosity 147. Four Important Concyclic Points
				Curiosity 148. Four Remarkable Concyclic Points
				Curiosity 149. Four Unexpected Concyclic Points
				Curiosity 150. Perpendiculars that Generate Concyclic Points
				Curiosity 151. Altitudes and Circles that Generate Another Circle
				Curiosity 152. More Unexpected Concyclic Points
				Curiosity 153. A Surprising Five-Point Circle
				Curiosity 154. The Famous Nine-Point Circle
				Curiosity 155. A Collinearity with the Center of the Nine-Point Circle
				Curiosity 156. The Meeting of the Three Famous Triangle Centers
				Curiosity 157. Properties of the Nine-Point Circle
				Curiosity 158. More Properties of the Nine-Point Circle
				Curiosity 159. An Unexpected Collinearity
				Curiosity 160. A Concurrency Generated by the Orthic Triangle
				Curiosity 161. Altitudes Produce a Concurrency and Equality
				Curiosity 162. Napoleon’s Contribution to Mathematics
				Curiosity 163. Napoleon’s Minimum Distance Point (the Fermat Point)
				Curiosity 164. When the Minimum Distance Point is not Inside the Triangle
				Curiosity 165. Extensions of Napoleon’s Theorem
				Curiosity 166. Overlapping Side-Equilateral Triangles
				Curiosity 167. Surprising Triangle Area Relationship
				Curiosity 168. The Centroid Enters the Previous Configuration
				Curiosity 169. The Emergence of Another Equilateral Triangle
				Curiosity 170. A Novel Way of Finding the Center of the Circumscribed Circle
				Curiosity 171. A Concurrency Point of Circles
				Curiosity 172. The Famous Miquel Theorem
				Curiosity 173. Miquel’s Similar Triangles
				Curiosity 174. The Astounding Morley’s Theorem
				Curiosity 175. Morley’s Theorem Extended
		Proofs of the Triangle Curiosities
			Curiosity 1. Angles at the Incenter
				Curiosity 2. An Unexpected Perpendicularity
				Curiosity 3. A Most Unusual Line Bisection
				Curiosity 4. Gergonne’s Discovery Involving the Inscribed Circle of a Triangle
				Curiosity 5. Another Unexpected Concurrency for Gergonne’s Triangle
				Curiosity 6. A Novelty Concerning the Circumscribed and Inscribed Circles
				Curiosity 7. Concurrency Involving the Inscribed Circle
				Curiosity 8. A More General Concurrency Involving the Inscribed Circle
				Curiosity 9. An Inscribed Circle of a General Triangle Generates Equal Angles
				Curiosity 10. Surprising Equal Angles
				Curiosity 11. An Inscribed Circle of a Triangle Generates its Circumscribed Circle
				Curiosity 12. The Feet of the Altitudes Partition a Triangle into Three Pairs of Equal-Area Triangles
				Curiosity 13. The Noteworthy Position of the Orthocenter of a Triangle
				Curiosity 14. A Circle Intersects a Triangle to Generate Lots of Equal Segments
				Curiosity 15. A Strange Appearance of a Congruent Triangle
				Curiosity 16. The Orthocenter Joins Three Other Points in Collinearity
				Curiosity 17. The Orthocenter Appears as the Midpoint of a Line Segment
				Curiosity 18. A Conglomeration of Perpendiculars Generates Equal Line Segments
				Curiosity 19. Yet Another Unexpected Collinearity
				Curiosity 20. Introducing the Orthic Triangle
				Curiosity 21. Finding the Perimeter of the Orthic Triangle
				Curiosity 22. The Orthic Triangle is a Triangle’s Smallest Inscribed Triangle
				Curiosity 23. The Orthic Triangle’s Surprising Similar Partner
				Curiosity 24. A Nice Concurrency Generated by the Orthic Triangle
				Curiosity 25. Orthic Triangle Generates an Isosceles Triangle
				Curiosity 26. The Orthic Triangle Generates a Parallelogram
				Curiosity 27. Concyclic Points Generated by the Orthic Triangle
				Curiosity 28. The Orthic Triangle Generates Concurrent Lines
				Curiosity 29. The Orthic Triangle Generates Collinear Points
				Curiosity 30. An Unexpected Property of Altitude Feet of a Triangle
				Curiosity 31. How a Non-Isosceles Triangle Can Generate a Parallelogram
				Curiosity 32. Perpendiculars Generating Unexpected Parallel Lines
				Curiosity 33. Tangents and Inscribed-Triangle Sides Generate a Surprising Collinearity
				Curiosity 34. Creating Congruent Triangles Inscribed in the Same Circle
				Curiosity 35. An Unexpected Product Equality
				Curiosity 36. An Unusual Product of Two Sides of a Triangle
				Curiosity 37. Simson’s Theorem
				Curiosity 38. An Extension of Simson’s Theorem
				Curiosity 39. An Interesting Aspect of Simson’s Theorem
				Curiosity 40. A Parallel to the Simson Line
				Curiosity 41. Two Triangles Related by a Common Point: Circumcenter – Centroid
				Curiosity 42. Introducing the Medians of a Triangle with Some of Their Amazing Properties
				Curiosity 43. The Median of a Triangle is Equidistant from Two Vertices
				Curiosity 44. The Special Median of a Right Triangle
				Curiosity 45. Medians Partition Any Triangle into Four Congruent Triangles
				Curiosity 46. How the Centroid Helps Create a Similar Triangle
				Curiosity 47. The Centroid can Provide a Most Unusual Balance
				Curiosity 48. Distances from a Triangle’s Vertices to a Random Line
				Curiosity 49. A Special Centroid Property When Two Medians are Perpendicular
				Curiosity 50. The Centroid’s Amazing Property
				Curiosity 51. Some Median Surprises
				Curiosity 52. More Median Marvels
				Curiosity 53. Comparing Medians to Triangle Perimeters
				Curiosity 54. Median Extensions Generate Collinearity
				Curiosity 55. Two Unusual Triangles Share a Common Centroid
				Curiosity 56. The Circumscribed Circle Revisited: The Incredible Relationship of the Centers of the Circumscribed Circles of the Median Triangles
				Curiosity 57. An Astonishing Equality
				Curiosity 58. An Unexpected Simultaneous Bisection and Quadrisection in a Triangle
				Curiosity 59. Noteworthy Triangle Area Relations
				Curiosity 60. A Special Feature of a Random Point in an Equilateral Triangle
				Curiosity 61. A Special Feature of a Point Outside of an Equilateral Triangle
				Curiosity 62. Using an Equilateral Triangle to Trisect a Line Segment
				Curiosity 63. Another Way to Trisect a Line Segment with an Equilateral Triangle
				Curiosity 64. An Unusual Construction of a 30°-60°- 90° Triangle inside an Equilateral Triangle
				Curiosity 65. An Unusual Product of Segments in an Equilateral Triangle
				Curiosity 66. A Surprising Relationship between an Isosceles Triangle and an Equilateral Triangle
				Curiosity 67. An Unexpected Equality in an Isosceles Triangle
				Curiosity 68. Another Unexpected Equality in an Isosceles Triangle
				Curiosity 69. A Remarkable Geometric Equality
				Curiosity 70. Another Remarkable Geometric Equality
				Curiosity 71. An Unexpected Perpendicularity
				Curiosity 72. The Unexpected Property of an Altitude to the Side of an Isosceles Triangle
				Curiosity 73. Peculiar Property of Isosceles Triangles
				Curiosity 74. A Bizarre Connection: The Triangle Incenter on its Circumcircle
				Curiosity 75. Collinear Points Generate an Angle Bisector
				Curiosity 76. Unexpected Similar Triangles
				Curiosity 77. Perpendiculars to Four Angle Bisectors of a Triangle Reveal Four Collinear Points
				Curiosity 78. A Convoluted Bisection of a Line Segment
				Curiosity 79. Determining the Perimeter of a Triangle Without Measuring its Side Lengths
				Curiosity 80. A Strange Angle Trisection
				Curiosity 81. A Most Unexpected Equality
				Curiosity 82. A Triangle Peculiarity
				Curiosity 83. A Counterintuitive Area Equality of Triangles
				Curiosity 84. Parallel Lines Create a Double Area Triangle
				Curiosity 85. Unexpected Triangle Area Relationships
				Curiosity 86. Creating a Triangle Whose Area is Three-Quarters the Area of a Given Triangle
				Curiosity 87. An Astounding Construction: Similar Triangles Whose Area Ratio is 1:4
				Curiosity 88. Unforeseen Equality of Inscribed Triangles
				Curiosity 89. The Unanticipated Commonality of Equal Area Triangles
				Curiosity 90. How a Random Point Divides the Area of a Triangle in Half
				Curiosity 91. Determining a Triangle One-Third of the Area of a Given Triangle
				Curiosity 92. Trisection Points Partitioning a Triangle
				Curiosity 93. Further Surprises Provided by the Trisection Points of Triangle Sides
				Curiosity 94. Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 95. Yet Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 96. Medians and Trisectors Partitioning a Triangle with an Unexpected Result
				Curiosity 97. The Unforeseen Characteristic of a Random Triangle with a 60° Angle
				Curiosity 98. Another Surprising Feature of a Random Triangle with a 60° Angle
				Curiosity 99. Another Unexpected Collinearity
				Curiosity 100. Reflections of Triangles Generate Concurrent Circles and Concurrent Lines
				Curiosity 101. The Wonders of Three Concurrent Congruent Circles
				Curiosity 102. Further Unexpected Concurrencies
				Curiosity 103. One Concurrency Generates Another Concurrency
				Curiosity 104. Unusual Perpendiculars Generating Concurrencies
				Curiosity 105. A Most Unexpected Concurrency
				Curiosity 106. An Intriguing Concurrency
				Curiosity 107. Another Unexpected and Unforeseen Concurrency
				Curiosity 108. A Most Unexpected Concurrency from a Triangle
				Curiosity 109. Astounding Point Property in a Triangle
				Curiosity 110. A Counterintuitive Concurrency
				Curiosity 111. Concurrent Angle Bisectors of Triangles with a Common Base
				Curiosity 112. An Unexpected Concurrency with the Circumscribed Circle
				Curiosity 113. Another Surprising Aspect of Concurrent Cevians
				Curiosity 114. A Surprising Equality
				Curiosity 115. An Unforeseen Triangle Surprise
				Curiosity 116. A Remarkable Property of Two Triangles with a Common Base
				Curiosity 117. A Surprising Line Partitioning
				Curiosity 118. The Hidden Length Equality of Line Segments in a Triangle
				Curiosity 119. The Unexpected Angle Measure
				Curiosity 120. An Unexpected Equality from a Right Triangle
				Curiosity 121. The Conundrum: Perpendicular or Parallel
				Curiosity 122. Multiple Midpoints in a Right Triangle Determine Equal Line Segments
				Curiosity 123. Perpendiculars in Right Triangles that Generate Equal Angles
				Curiosity 124. Right Triangles Sharing a Common Hypotenuse Generate an Unexpected Equality
				Curiosity 125. The Pythagorean Theorem Revisited Geometrically
				Curiosity 126. Squares on Triangle Sides Produce Noteworthy Surprises
				Curiosity 127. More Properties Generated by Squares on Triangle Sides
				Curiosity 128. An Unexpected Perpendicularity
				Curiosity 129. A Surprising Concurrency from Squares on the Sides of a Triangle
				Curiosity 130. Another Surprising Concurrency from Squares on Triangle Sides
				Curiosity 131. Using Squares on Triangle Sides to Create another Surprising Concurrency
				Curiosity 132. Squares on the Legs of a Right Triangle Produce an Unexpected Equality
				Curiosity 133. More About Squares on the Legs of a Right Triangle
				Curiosity 134. More Placements of Squares on Right Triangles
				Curiosity 135. Another Unexpected Area Equality
				Curiosity 136. The Square on the Hypotenuse of a Right Triangle
				Curiosity 137. A Truly Unexpected Collinearity
				Curiosity 138. A Most Unusual Procedure to Divide a Square into Two Equal Parts
				Curiosity 139. Doubling a Square
				Curiosity 140. A Strange Construction of Parallel Lines
				Curiosity 141. An Unusual Technique to Find the Midpoint of a Line Segment
				Curiosity 142. The Unexpected Appearance of Parallel Lines
				Curiosity 143. The Unanticipated Parallel Line
				Curiosity 144. The Surprising Perpendicularity
				Curiosity 145. Another Unexpected Perpendicularity
				Curiosity 146. Yet Another Unexpected Right Angle
				Curiosity 147. Four Important Concyclic Points
				Curiosity 148. Four Remarkable Concyclic Points
				Curiosity 149. Four Unexpected Concyclic Points
				Curiosity 150. Perpendiculars that Generate Concyclic Points
				Curiosity 151. Altitudes and Circles that Generate Another Circle
				Curiosity 152. More Unexpected Concyclic Points
				Curiosity 153. A Surprising Five-Point Circle
				Curiosity 154. The Famous Nine-Point Circle
				Curiosity 155. A Collinearity with the Center of the Nine-Point Circle
				Curiosity 156. The Meeting of the Three Famous Triangle Centers
				Curiosity 157. Properties of the Nine-Point Circle
				Curiosity 158. More Properties of the Nine-Point Circle
				Curiosity 159. An Unexpected Collinearity
				Curiosity 160. A Concurrency Generated by the Orthic Triangle
				Curiosity 161. Altitudes Produce a Concurrency and Equality
				Curiosity 162. Napoleon’s Contribution to Mathematics
				Curiosity 163. Napoleon’s Minimum Distance Point (The Fermat Point)
				Curiosity 164. When the Minimum Distance Point is not Inside the Triangle
				Curiosity 165. Extensions of Napoleon’s Theorem
				Curiosity 166. Overlapping Side-Equilateral Triangles
				Curiosity 167. Surprising Triangle Area Relationship
				Curiosity 168. The Centroid Enters the Previous Configuration
				Curiosity 169. The Emergence of Another Equilateral Triangle
				Curiosity 170. A Novel Way of Finding the Center of the Circumscribed Circle
				Curiosity 171. A Concurrency Point of Circles
				Curiosity 172. The Famous Miquel Theorem
				Curiosity 173. Miquel’s Similar Triangles
				Curiosity 174. The Astounding Morley’s Theorem
				Curiosity 175. Morley’s Theorem Extended
		Toolbox
			Introduction: The Geometry Toolbox
			A.	Tools You Are Probably Familiar with from the High School Geometry Course
				A1: Congruence of Triangles
				A2: Similarity of Triangles
				A3: Right Triangle Properties (See Figure A3)
				A4: Angles Related to a Circle (see Figure A4)
				A5:	Tangents, Secants, and Chords: Segments of a Circle (see Figure A5)
				A6: The Law of Sines and the Law of Cosines
				A7: Angle Sum and Difference Identities
			B.	Less Familiar Tools—However, Useful and Fascinating
				B1. Interior Angle Bisector in a Triangle
				B2. Exterior Angle Bisector in a Triangle
				B3. Menelaus’ Theorem
				B4. Ceva’s Theorem
				B5. The Trigonometric Version of Ceva’s Theorem
				B6. Ceva’s Theorem Extended
				B7. Desargues’s Theorem
				B8. Stewart’s Theorem
				B9. Theorem of Apollonius
				B10. Ptolemy’s Theorem
				B11. Isometries: Reflection and Rotation
				B12. Homothety and Similarity
				B13. Polarity on Circles
	Index
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	Contents
	Acknowledgments
	About the Authors
		Introduction
		Geometric Curiosities: Introducing the Triangle
			Curiosity 1. Angles at the Incenter
				Curiosity 2. An Unexpected Perpendicularity
				Curiosity 3. A Most Unusual Line Bisection
				Curiosity 4. Gergonne’s Discovery Involving the Inscribed Circle of a Triangle
				Curiosity 5. Another Unexpected Concurrency for Gergonne’s Triangle
				Curiosity 6. A Novelty Concerning the Circumscribed and Inscribed Circles
				Curiosity 7. Concurrency Involving the Inscribed Circle
				Curiosity 8. A More General Concurrency Involving the Inscribed Circle
				Curiosity 9. An Inscribed Circle of a General Triangle Generates Equal Angles
				Curiosity 10. Surprising Equal Angles
				Curiosity 11. An Inscribed Circle of a Triangle Generates its Circumscribed Circle
				Curiosity 12. The Feet of the Altitudes Partition a Triangle into Three Pairs of Equal-Area Triangles
				Curiosity 13. The Noteworthy Position of the Orthocenter of a Triangle
				Curiosity 14. A Circle Intersects a Triangle to Generate Lots of Equal Segments
				Curiosity 15. A Strange Appearance of a Congruent Triangle
				Curiosity 16. The Orthocenter Joins Three Other Points in Collinearity
				Curiosity 17. The Orthocenter Appears as the Midpoint of a Line Segment
				Curiosity 18. A Conglomeration of Perpendiculars Generates Equal Line Segments
				Curiosity 19. Yet Another Unexpected Collinearity
				Curiosity 20. Introducing the Orthic Triangle
				Curiosity 21. Finding the Perimeter of the Orthic Triangle
				Curiosity 22. The Orthic Triangle is a Triangle’s Smallest Inscribed Triangle
				Curiosity 23. The Orthic Triangle’s Surprising Similar Partner
				Curiosity 24. A Nice Concurrency Generated by the Orthic Triangle
				Curiosity 25. Orthic Triangle Generates an Isosceles Triangle
				Curiosity 26. The Orthic Triangle Generates a Parallelogram
				Curiosity 27. Concyclic Points Generated by the Orthic Triangle
				Curiosity 28. The Orthic Triangle Generates Concurrent Lines
				Curiosity 29. The Orthic Triangle Generates Collinear Points
				Curiosity 30. An Unexpected Property of Altitude Feet of a Triangle
				Curiosity 31. How a Non-Isosceles Triangle Can Generate a Parallelogram
				Curiosity 32. Perpendiculars Generating Unexpected Parallel Lines
				Curiosity 33. Inscribed Triangle Generates Collinearity
				Curiosity 34. Unexpected Congruent Triangles in a Circle
				Curiosity 35. An Unexpected Product Equality
				Curiosity 36. An Unusual Product of Two Sides of a Triangle
				Curiosity 37. Simson’s Theorem
				Curiosity 38. An Extension of Simson’s Theorem
				Curiosity 39. An Interesting Aspect of Simson’s Theorem
				Curiosity 40. A Parallel to the Simson Line
				Curiosity 41. Two Triangles Related by a Common Point: Circumcenter – Centroid
				Curiosity 42. Introducing the Medians of a Triangle with Some of Their Amazing Properties
				Curiosity 43. The Median of a Triangle is Equidistant from Two Vertices
				Curiosity 44. The Special Median of a Right Triangle
				Curiosity 45. Medians Partition Any Triangle into Four Congruent Triangles
				Curiosity 46. How the Centroid Helps Create a Similar Triangle
				Curiosity 47. The Centroid can Provide a Most Unusual Balance
				Curiosity 48. Distances from a Triangle’s Vertices to a Random Line
				Curiosity 49. A Special Centroid Property When Two Medians are Perpendicular
				Curiosity 50. The Centroid’s Amazing Property
				Curiosity 51. Some Median Surprises
				Curiosity 52. More Median Marvels
				Curiosity 53. Comparing Medians to Triangle Perimeters
				Curiosity 54. Median Extensions Generate Collinearity
				Curiosity 55. Two Unusual Triangles Share a Common Centroid
				Curiosity 56. The Circumscribed Circle Revisited: The Incredible Relationship of the Centers of the Circumscribed Circles of the Median Triangles
				Curiosity 57. An Astonishing Equality
				Curiosity 58. An Unexpected Simultaneous Bisection and Quadrisection in a Triangle
				Curiosity 59. Noteworthy Triangle Area Relations
				Curiosity 60. A Special Feature of a Random Point in an Equilateral Triangle
				Curiosity 61. A Special Feature of a Point Outside of an Equilateral Triangle
				Curiosity 62. Using an Equilateral Triangle to Trisect a Line Segment
				Curiosity 63. Another Way to Trisect a Line Segment with an Equilateral Triangle
				Curiosity 64. An Unusual Construction of a 30°-60°-90° Triangle Inside an Equilateral Triangle
				Curiosity 65. An Unusual Product of Segments in an Equilateral Triangle
				Curiosity 66. A Surprising Relationship Between an Isosceles Triangle and an Equilateral Triangle
				Curiosity 67. An Unexpected Equality in an Isosceles Triangle
				Curiosity 68. Another Unexpected Equality in an Isosceles Triangle
				Curiosity 69. A Remarkable Geometric Equality
				Curiosity 70. Another Remarkable Geometric Equality
				Curiosity 71. An Unexpected Perpendicularity
				Curiosity 72. The Unexpected Property of an Altitude to the Side of an Isosceles Triangle
				Curiosity 73. Peculiar Property of Isosceles Triangles
				Curiosity 74. A Bizarre Connection: The Triangle Incenter on its Circumcircle
				Curiosity 75. Collinear Points Generate an Angle Bisector
				Curiosity 76. Unexpected Similar Triangles
				Curiosity 77. Perpendiculars to Four Angle Bisectors of a Triangle Reveal Four Collinear Points
				Curiosity 78. A Convoluted Bisection of a Line Segment
				Curiosity 79. Determining the Perimeter of a Triangle Without Measuring its Side Lengths
				Curiosity 80. A Strange Angle Trisection
				Curiosity 81. A Most Unexpected Equality
				Curiosity 82. A Triangle Peculiarity
				Curiosity 83. A Counterintuitive Area Equality of Triangles
				Curiosity 84. Parallel Lines Create a Double Area Triangle
				Curiosity 85. Unexpected Triangle Area Relationships
				Curiosity 86. Creating a Triangle Whose Area is Three-Quarters the Area of a Given Triangle
				Curiosity 87. An Astounding Construction: Similar Triangles Whose Area Ratio is 1:4
				Curiosity 88. Unforeseen Equality of Inscribed Triangles
				Curiosity 89. The Unanticipated Commonality of Equal Area Triangles
				Curiosity 90. How a Random Point Divides the Area of a Triangle in Half
				Curiosity 91. Determining a Triangle One-Third of the Area of a Given Triangle
				Curiosity 92. Trisection Points Partitioning a Triangle
				Curiosity 93. Further Surprises Provided by the Trisection Points of Triangle Sides
				Curiosity 94. Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 95. Yet Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 96. Medians and Trisectors Partitioning a Triangle with an Unexpected Result
				Curiosity 97. The Unforeseen Characteristic of a Random Triangle with a 60° Angle
				Curiosity 98. Another Surprising Feature of a Random Triangle with a 60° Angle
				Curiosity 99. Another Unexpected Collinearity
				Curiosity 100. Reflections of Triangles Generate Concurrent Circles and Concurrent Lines
				Curiosity 101. The Wonders of Three Concurrent Congruent Circles
				Curiosity 102. Further Unexpected Concurrencies
				Curiosity 103. One Concurrency Generates Another Concurrency
				Curiosity 104. Unusual Perpendiculars Generating Concurrencies
				Curiosity 105. A Most Unexpected Concurrency
				Curiosity 106. An Intriguing Concurrency
				Curiosity 107. Another Unexpected and Unforeseen Concurrency
				Curiosity 108. A Most Unexpected Concurrency from a Triangle
				Curiosity 109. Astounding Point Property in a Triangle
				Curiosity 110. A Counterintuitive Concurrency
				Curiosity 111. Concurrent Angle Bisectors of Triangles with a Common Base
				Curiosity 112. An Unexpected Concurrency with the Circumscribed Circle
				Curiosity 113. Another Surprising Aspect of Concurrent Cevians
				Curiosity 114. A Surprising Equality
				Curiosity 115. An Unforeseen Triangle Surprise
				Curiosity 116. A Remarkable Property of Two Triangles with a Common Base
				Curiosity 117. A Surprising Line Partitioning
				Curiosity 118. The Hidden Length Equality of Line Segments in a Triangle
				Curiosity 119. The Unexpected Angle Measure
				Curiosity 120. An Unexpected Equality from a Right Triangle
				Curiosity 121. The Conundrum: Perpendicular or Parallel
				Curiosity 122. Multiple Midpoints in a Right Triangle Determine Equal Line Segments
				Curiosity 123. Perpendiculars in Right Triangles that Generate Equal Angles
				Curiosity 124. Right Triangles Sharing a Common Hypotenuse Generate an Unexpected Equality
				Curiosity 125. The Pythagorean Theorem Revisited Geometrically
				Curiosity 126. Squares on Triangle Sides Produce Noteworthy Surprises
				Curiosity 127. More Properties Generated by Squares on Triangle Sides
				Curiosity 128. An Unexpected Perpendicularity
				Curiosity 129. A Surprising Concurrency from Squares on the Sides of a Triangle
				Curiosity 130. Another Surprising Concurrency from Squares on Triangle Sides
				Curiosity 131. Using Squares on Triangle Sides to Create another Surprising Concurrency
				Curiosity 132. Squares on the Legs of a Right Triangle Produce an Unexpected Equality
				Curiosity 133. More About Squares on the Legs of a Right Triangle
				Curiosity 134. More Placements of Squares on Right Triangles
				Curiosity 135. Another Unexpected Area Equality
				Curiosity 136. The Square on the Hypotenuse of a Right Triangle
				Curiosity 137. A Truly Unexpected Collinearity
				Curiosity 138. A Most Unusual Procedure to Divide a Square into Two Equal Parts
				Curiosity 139. Doubling a Square
				Curiosity 140. A Strange Construction of Parallel Lines
				Curiosity 141. An Unusual Technique to Find the Midpoint of a Line Segment
				Curiosity 142. The Unexpected Appearance of Parallel Lines
				Curiosity 143. The Unanticipated Parallel Line
				Curiosity 144. The Surprising Perpendicularity
				Curiosity 145. Another Unexpected Perpendicularity
				Curiosity 146. Yet Another Unexpected Right Angle
				Curiosity 147. Four Important Concyclic Points
				Curiosity 148. Four Remarkable Concyclic Points
				Curiosity 149. Four Unexpected Concyclic Points
				Curiosity 150. Perpendiculars that Generate Concyclic Points
				Curiosity 151. Altitudes and Circles that Generate Another Circle
				Curiosity 152. More Unexpected Concyclic Points
				Curiosity 153. A Surprising Five-Point Circle
				Curiosity 154. The Famous Nine-Point Circle
				Curiosity 155. A Collinearity with the Center of the Nine-Point Circle
				Curiosity 156. The Meeting of the Three Famous Triangle Centers
				Curiosity 157. Properties of the Nine-Point Circle
				Curiosity 158. More Properties of the Nine-Point Circle
				Curiosity 159. An Unexpected Collinearity
				Curiosity 160. A Concurrency Generated by the Orthic Triangle
				Curiosity 161. Altitudes Produce a Concurrency and Equality
				Curiosity 162. Napoleon’s Contribution to Mathematics
				Curiosity 163. Napoleon’s Minimum Distance Point (the Fermat Point)
				Curiosity 164. When the Minimum Distance Point is not Inside the Triangle
				Curiosity 165. Extensions of Napoleon’s Theorem
				Curiosity 166. Overlapping Side-Equilateral Triangles
				Curiosity 167. Surprising Triangle Area Relationship
				Curiosity 168. The Centroid Enters the Previous Configuration
				Curiosity 169. The Emergence of Another Equilateral Triangle
				Curiosity 170. A Novel Way of Finding the Center of the Circumscribed Circle
				Curiosity 171. A Concurrency Point of Circles
				Curiosity 172. The Famous Miquel Theorem
				Curiosity 173. Miquel’s Similar Triangles
				Curiosity 174. The Astounding Morley’s Theorem
				Curiosity 175. Morley’s Theorem Extended
		Proofs of the Triangle Curiosities
			Curiosity 1. Angles at the Incenter
				Curiosity 2. An Unexpected Perpendicularity
				Curiosity 3. A Most Unusual Line Bisection
				Curiosity 4. Gergonne’s Discovery Involving the Inscribed Circle of a Triangle
				Curiosity 5. Another Unexpected Concurrency for Gergonne’s Triangle
				Curiosity 6. A Novelty Concerning the Circumscribed and Inscribed Circles
				Curiosity 7. Concurrency Involving the Inscribed Circle
				Curiosity 8. A More General Concurrency Involving the Inscribed Circle
				Curiosity 9. An Inscribed Circle of a General Triangle Generates Equal Angles
				Curiosity 10. Surprising Equal Angles
				Curiosity 11. An Inscribed Circle of a Triangle Generates its Circumscribed Circle
				Curiosity 12. The Feet of the Altitudes Partition a Triangle into Three Pairs of Equal-Area Triangles
				Curiosity 13. The Noteworthy Position of the Orthocenter of a Triangle
				Curiosity 14. A Circle Intersects a Triangle to Generate Lots of Equal Segments
				Curiosity 15. A Strange Appearance of a Congruent Triangle
				Curiosity 16. The Orthocenter Joins Three Other Points in Collinearity
				Curiosity 17. The Orthocenter Appears as the Midpoint of a Line Segment
				Curiosity 18. A Conglomeration of Perpendiculars Generates Equal Line Segments
				Curiosity 19. Yet Another Unexpected Collinearity
				Curiosity 20. Introducing the Orthic Triangle
				Curiosity 21. Finding the Perimeter of the Orthic Triangle
				Curiosity 22. The Orthic Triangle is a Triangle’s Smallest Inscribed Triangle
				Curiosity 23. The Orthic Triangle’s Surprising Similar Partner
				Curiosity 24. A Nice Concurrency Generated by the Orthic Triangle
				Curiosity 25. Orthic Triangle Generates an Isosceles Triangle
				Curiosity 26. The Orthic Triangle Generates a Parallelogram
				Curiosity 27. Concyclic Points Generated by the Orthic Triangle
				Curiosity 28. The Orthic Triangle Generates Concurrent Lines
				Curiosity 29. The Orthic Triangle Generates Collinear Points
				Curiosity 30. An Unexpected Property of Altitude Feet of a Triangle
				Curiosity 31. How a Non-Isosceles Triangle Can Generate a Parallelogram
				Curiosity 32. Perpendiculars Generating Unexpected Parallel Lines
				Curiosity 33. Tangents and Inscribed-Triangle Sides Generate a Surprising Collinearity
				Curiosity 34. Creating Congruent Triangles Inscribed in the Same Circle
				Curiosity 35. An Unexpected Product Equality
				Curiosity 36. An Unusual Product of Two Sides of a Triangle
				Curiosity 37. Simson’s Theorem
				Curiosity 38. An Extension of Simson’s Theorem
				Curiosity 39. An Interesting Aspect of Simson’s Theorem
				Curiosity 40. A Parallel to the Simson Line
				Curiosity 41. Two Triangles Related by a Common Point: Circumcenter – Centroid
				Curiosity 42. Introducing the Medians of a Triangle with Some of Their Amazing Properties
				Curiosity 43. The Median of a Triangle is Equidistant from Two Vertices
				Curiosity 44. The Special Median of a Right Triangle
				Curiosity 45. Medians Partition Any Triangle into Four Congruent Triangles
				Curiosity 46. How the Centroid Helps Create a Similar Triangle
				Curiosity 47. The Centroid can Provide a Most Unusual Balance
				Curiosity 48. Distances from a Triangle’s Vertices to a Random Line
				Curiosity 49. A Special Centroid Property When Two Medians are Perpendicular
				Curiosity 50. The Centroid’s Amazing Property
				Curiosity 51. Some Median Surprises
				Curiosity 52. More Median Marvels
				Curiosity 53. Comparing Medians to Triangle Perimeters
				Curiosity 54. Median Extensions Generate Collinearity
				Curiosity 55. Two Unusual Triangles Share a Common Centroid
				Curiosity 56. The Circumscribed Circle Revisited: The Incredible Relationship of the Centers of the Circumscribed Circles of the Median Triangles
				Curiosity 57. An Astonishing Equality
				Curiosity 58. An Unexpected Simultaneous Bisection and Quadrisection in a Triangle
				Curiosity 59. Noteworthy Triangle Area Relations
				Curiosity 60. A Special Feature of a Random Point in an Equilateral Triangle
				Curiosity 61. A Special Feature of a Point Outside of an Equilateral Triangle
				Curiosity 62. Using an Equilateral Triangle to Trisect a Line Segment
				Curiosity 63. Another Way to Trisect a Line Segment with an Equilateral Triangle
				Curiosity 64. An Unusual Construction of a 30°-60°- 90° Triangle inside an Equilateral Triangle
				Curiosity 65. An Unusual Product of Segments in an Equilateral Triangle
				Curiosity 66. A Surprising Relationship between an Isosceles Triangle and an Equilateral Triangle
				Curiosity 67. An Unexpected Equality in an Isosceles Triangle
				Curiosity 68. Another Unexpected Equality in an Isosceles Triangle
				Curiosity 69. A Remarkable Geometric Equality
				Curiosity 70. Another Remarkable Geometric Equality
				Curiosity 71. An Unexpected Perpendicularity
				Curiosity 72. The Unexpected Property of an Altitude to the Side of an Isosceles Triangle
				Curiosity 73. Peculiar Property of Isosceles Triangles
				Curiosity 74. A Bizarre Connection: The Triangle Incenter on its Circumcircle
				Curiosity 75. Collinear Points Generate an Angle Bisector
				Curiosity 76. Unexpected Similar Triangles
				Curiosity 77. Perpendiculars to Four Angle Bisectors of a Triangle Reveal Four Collinear Points
				Curiosity 78. A Convoluted Bisection of a Line Segment
				Curiosity 79. Determining the Perimeter of a Triangle Without Measuring its Side Lengths
				Curiosity 80. A Strange Angle Trisection
				Curiosity 81. A Most Unexpected Equality
				Curiosity 82. A Triangle Peculiarity
				Curiosity 83. A Counterintuitive Area Equality of Triangles
				Curiosity 84. Parallel Lines Create a Double Area Triangle
				Curiosity 85. Unexpected Triangle Area Relationships
				Curiosity 86. Creating a Triangle Whose Area is Three-Quarters the Area of a Given Triangle
				Curiosity 87. An Astounding Construction: Similar Triangles Whose Area Ratio is 1:4
				Curiosity 88. Unforeseen Equality of Inscribed Triangles
				Curiosity 89. The Unanticipated Commonality of Equal Area Triangles
				Curiosity 90. How a Random Point Divides the Area of a Triangle in Half
				Curiosity 91. Determining a Triangle One-Third of the Area of a Given Triangle
				Curiosity 92. Trisection Points Partitioning a Triangle
				Curiosity 93. Further Surprises Provided by the Trisection Points of Triangle Sides
				Curiosity 94. Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 95. Yet Another Surprise Provided by the Trisection Points of Triangle Sides
				Curiosity 96. Medians and Trisectors Partitioning a Triangle with an Unexpected Result
				Curiosity 97. The Unforeseen Characteristic of a Random Triangle with a 60° Angle
				Curiosity 98. Another Surprising Feature of a Random Triangle with a 60° Angle
				Curiosity 99. Another Unexpected Collinearity
				Curiosity 100. Reflections of Triangles Generate Concurrent Circles and Concurrent Lines
				Curiosity 101. The Wonders of Three Concurrent Congruent Circles
				Curiosity 102. Further Unexpected Concurrencies
				Curiosity 103. One Concurrency Generates Another Concurrency
				Curiosity 104. Unusual Perpendiculars Generating Concurrencies
				Curiosity 105. A Most Unexpected Concurrency
				Curiosity 106. An Intriguing Concurrency
				Curiosity 107. Another Unexpected and Unforeseen Concurrency
				Curiosity 108. A Most Unexpected Concurrency from a Triangle
				Curiosity 109. Astounding Point Property in a Triangle
				Curiosity 110. A Counterintuitive Concurrency
				Curiosity 111. Concurrent Angle Bisectors of Triangles with a Common Base
				Curiosity 112. An Unexpected Concurrency with the Circumscribed Circle
				Curiosity 113. Another Surprising Aspect of Concurrent Cevians
				Curiosity 114. A Surprising Equality
				Curiosity 115. An Unforeseen Triangle Surprise
				Curiosity 116. A Remarkable Property of Two Triangles with a Common Base
				Curiosity 117. A Surprising Line Partitioning
				Curiosity 118. The Hidden Length Equality of Line Segments in a Triangle
				Curiosity 119. The Unexpected Angle Measure
				Curiosity 120. An Unexpected Equality from a Right Triangle
				Curiosity 121. The Conundrum: Perpendicular or Parallel
				Curiosity 122. Multiple Midpoints in a Right Triangle Determine Equal Line Segments
				Curiosity 123. Perpendiculars in Right Triangles that Generate Equal Angles
				Curiosity 124. Right Triangles Sharing a Common Hypotenuse Generate an Unexpected Equality
				Curiosity 125. The Pythagorean Theorem Revisited Geometrically
				Curiosity 126. Squares on Triangle Sides Produce Noteworthy Surprises
				Curiosity 127. More Properties Generated by Squares on Triangle Sides
				Curiosity 128. An Unexpected Perpendicularity
				Curiosity 129. A Surprising Concurrency from Squares on the Sides of a Triangle
				Curiosity 130. Another Surprising Concurrency from Squares on Triangle Sides
				Curiosity 131. Using Squares on Triangle Sides to Create another Surprising Concurrency
				Curiosity 132. Squares on the Legs of a Right Triangle Produce an Unexpected Equality
				Curiosity 133. More About Squares on the Legs of a Right Triangle
				Curiosity 134. More Placements of Squares on Right Triangles
				Curiosity 135. Another Unexpected Area Equality
				Curiosity 136. The Square on the Hypotenuse of a Right Triangle
				Curiosity 137. A Truly Unexpected Collinearity
				Curiosity 138. A Most Unusual Procedure to Divide a Square into Two Equal Parts
				Curiosity 139. Doubling a Square
				Curiosity 140. A Strange Construction of Parallel Lines
				Curiosity 141. An Unusual Technique to Find the Midpoint of a Line Segment
				Curiosity 142. The Unexpected Appearance of Parallel Lines
				Curiosity 143. The Unanticipated Parallel Line
				Curiosity 144. The Surprising Perpendicularity
				Curiosity 145. Another Unexpected Perpendicularity
				Curiosity 146. Yet Another Unexpected Right Angle
				Curiosity 147. Four Important Concyclic Points
				Curiosity 148. Four Remarkable Concyclic Points
				Curiosity 149. Four Unexpected Concyclic Points
				Curiosity 150. Perpendiculars that Generate Concyclic Points
				Curiosity 151. Altitudes and Circles that Generate Another Circle
				Curiosity 152. More Unexpected Concyclic Points
				Curiosity 153. A Surprising Five-Point Circle
				Curiosity 154. The Famous Nine-Point Circle
				Curiosity 155. A Collinearity with the Center of the Nine-Point Circle
				Curiosity 156. The Meeting of the Three Famous Triangle Centers
				Curiosity 157. Properties of the Nine-Point Circle
				Curiosity 158. More Properties of the Nine-Point Circle
				Curiosity 159. An Unexpected Collinearity
				Curiosity 160. A Concurrency Generated by the Orthic Triangle
				Curiosity 161. Altitudes Produce a Concurrency and Equality
				Curiosity 162. Napoleon’s Contribution to Mathematics
				Curiosity 163. Napoleon’s Minimum Distance Point (The Fermat Point)
				Curiosity 164. When the Minimum Distance Point is not Inside the Triangle
				Curiosity 165. Extensions of Napoleon’s Theorem
				Curiosity 166. Overlapping Side-Equilateral Triangles
				Curiosity 167. Surprising Triangle Area Relationship
				Curiosity 168. The Centroid Enters the Previous Configuration
				Curiosity 169. The Emergence of Another Equilateral Triangle
				Curiosity 170. A Novel Way of Finding the Center of the Circumscribed Circle
				Curiosity 171. A Concurrency Point of Circles
				Curiosity 172. The Famous Miquel Theorem
				Curiosity 173. Miquel’s Similar Triangles
				Curiosity 174. The Astounding Morley’s Theorem
				Curiosity 175. Morley’s Theorem Extended
		Toolbox
			Introduction: The Geometry Toolbox
			A.	Tools You Are Probably Familiar with from the High School Geometry Course
				A1: Congruence of Triangles
				A2: Similarity of Triangles
				A3: Right Triangle Properties (See Figure A3)
				A4: Angles Related to a Circle (see Figure A4)
				A5:	Tangents, Secants, and Chords: Segments of a Circle (see Figure A5)
				A6: The Law of Sines and the Law of Cosines
				A7: Angle Sum and Difference Identities
			B.	Less Familiar Tools—However, Useful and Fascinating
				B1. Interior Angle Bisector in a Triangle
				B2. Exterior Angle Bisector in a Triangle
				B3. Menelaus’ Theorem
				B4. Ceva’s Theorem
				B5. The Trigonometric Version of Ceva’s Theorem
				B6. Ceva’s Theorem Extended
				B7. Desargues’s Theorem
				B8. Stewart’s Theorem
				B9. Theorem of Apollonius
				B10. Ptolemy’s Theorem
				B11. Isometries: Reflection and Rotation
				B12. Homothety and Similarity
				B13. Polarity on Circles
	Index