Geometric Gems: An Appreciation for Geometric Curiosities: Volume III: The Wonders of Circles

Contents
About the Authors
Acknowledgments
Introduction
Circle Curiosities: Introducing the Circle
	Curiosity 1. A Triangle Defines a Circle
	Curiosity 2. A Circle Tangent is Perpendicular to the Radius in the Point of Tangency
	Curiosity 3. Circle Tangents from a Common Point are of Equal Length
	Curiosity 4. The Angle Between Circle Tangents from a Common Point is Supplementary to its Intercepted Arc
	Curiosity 5. Angle Between a Circle Tangent and a Chord
	Curiosity 6. The Angle Formed by Two Chords of a Circle
	Curiosity 7. Surprising Supplementary Angles
	Curiosity 8. The Angle Formed by a Tangent and a Secant
	Curiosity 9. The Angle Formed by Two Secants
	Curiosity 10. Trisecting an Angle Using a Circle
	Curiosity 11. A Rather Surprising Equality
	Curiosity 12. A Strange Appearance of Parallel Lines
	Curiosity 13. Unexpected Complementary Angles in a Circle
	Curiosity 14. Surprising Squares Equality in a Circle
	Curiosity 15. A Surprise of a Circle Presenting a Familiar Triangle
	Curiosity 16. A Surprising Perpendicularity
	Curiosity 17. Unexpected Concyclic Points
	Curiosity 18. Circle Chords Generate a Constant Angle Sum
	Curiosity 19. A Circle Helps Divide a Line Segment into Two Equal Parts
	Curiosity 20. The Famous Butterfly Relationship
	Curiosity 21. A Circle Unexpectedly Produces Parallel Lines
	Curiosity 22. The Unexpected Constant for Any Point on a Circle
	Curiosity 23. Circle Tangents Generate Perpendicular Lines and Proportional Segments
	Curiosity 24. An Unexpected Circle Diameter
	Curiosity 25. An Unexpected Rectangle in a Circle
	Curiosity 26. Parallel Angle Bisectors
	Curiosity 27. Noteworthy Perpendiculars Generated by a Semicircle
	Curiosity 28. An Unexpected Angle Bisector in a Semicircle
	Curiosity 29. Tangent Circles on a Semicircle
	Curiosity 30. A Point on the Circumcircle of an Isosceles Triangle
	Curiosity 31. A Point on the Circumcircle of a Square
	Curiosity 32. A Point on the Circumcircle of a Regular Pentagon
	Curiosity 33. A Point on the Circumcircle of a Regular Hexagon
	Curiosity 34. The Tangent to the Circumscribed Circle of an Equilateral Triangle
	Curiosity 35. Surprising Concyclic Points
	Curiosity 36. The Midpoints of the Arcs of a Circle Produce Unexpected Parallels
	Curiosity 37. The Circumscribed Circle and the Altitudes of a Triangle Reveal an Unusual Equality
	Curiosity 38. The Unexpected Relationship Between the Midpoint of the Side of an Acute Triangle, its Circumcenter, and its Orthocenter
	Curiosity 39. An Unexpected Concurrency with the Circumscribed Circle of the Triangle
	Curiosity 40. An Unexpected Angle Bisector Hidden Within the Circumscribed Circle
	Curiosity 41. Surprising Concyclic Points Generated by a Triangle and its Circumscribed Circle
	Curiosity 42. Unexpected Concyclic Points
	Curiosity 43. The Orthocenter and Two Vertices of a Triangle Generate a Circle Equal to the Circumcircle
	Curiosity 44. The Appearance of an Unexpected Isosceles Triangle
	Curiosity 45. Perplexing Parallels from Concyclic Points
	Curiosity 46. Tangents to the Circumcircle of a Triangle
	Curiosity 47. Tangents to the Nine-Point Circle
	Curiosity 48. An Adventure with the Inscribed Circle of a Triangle and a Special Circumcircle
	Curiosity 49. Adventures with Circumscribed and Inscribed Circles of a Triangle
	Curiosity 50. More Adventures with Circumscribed and Inscribed Circles of a Triangle
	Curiosity 51. How Four Centers of a Triangle can Generate an Unusual Cyclic Quadrilateral
	Curiosity 52. The Radii of an Inscribed and Escribed Circle Determine a Rectangle Area
	Curiosity 53. A Circumscribed Circle and an Inscribed Circle Generate Three Equal-Length Lines
	Curiosity 54. Three Circles with a Surprising Common Point
	Curiosity 55. The Famous Conway Circle
	Curiosity 56. A Surprising Collinearity with the Incenter of a Triangle — Sawayama’s Lemma
	Curiosity 57. The Surprising Relationship Between the Four Tangent Circles of a Triangle
	Curiosity 58. Two Tangent Circles Generate an Unexpected Right Angle
	Curiosity 59. The Unexpected Diameter of a Circle
	Curiosity 60. Doubling the Distance Between Centers
	Curiosity 61. Tangent Circles Generating a Circle Tangent to the Line of Centers
	Curiosity 62. An Unexpected Angle Bisector Generated by Externally Tangent Circles
	Curiosity 63. Unexpected Parallel Chords in Externally Tangent Circles
	Curiosity 64. Surprising Parallel Tangents of Externally Tangent Circles
	Curiosity 65. Two Externally Tangent Circles Generate a Line Half as Long as Their Line of Centers
	Curiosity 66. A Very Surprising Constant Tangent Length
	Curiosity 67. Two Circles Creating Parallel Lines Unexpectedly
	Curiosity 68. Two Circles Create an Angle Bisector
	Curiosity 69. Tangent Circles Produce Unusual Line Products
	Curiosity 70. Equal Angles Appear in Internally Tangent Circles
	Curiosity 71. Incredible Tangent Lengths in Tangent Circles
	Curiosity 72. Symmetric Equal Angles
	Curiosity 73. An Amazing Rolling Result
	Curiosity 74. Semicircles — The Arbelos
	Curiosity 75. An Arbelos Bisection
	Curiosity 76. More on the Common Arbelos Bisection Point
	Curiosity 77. Yet More on the Arbelos
	Curiosity 78. Another Arbelos Surprise
	Curiosity 79. Some Further Entertainment with the Arbelos
	Curiosity 80. A Surprising Area Equality Created by Semicircles
	Curiosity 81. Another Surprising Area Equality Created by Semicircles
	Curiosity 82. An Amazing Curve Division into Two Equal Parts
	Curiosity 83. An Unexpected Bisection
	Curiosity 84. A Circle-Related Point with a Double Function
	Curiosity 85. Three Circles Through Two Common Points Generate a Surprising Circle
	Curiosity 86. Intersecting Circles Create Five Concyclic Points
	Curiosity 87. Concyclic Centers of Three Related Circumcircles
	Curiosity 88. Two Unequal Circles Counterintuitively Produce Two Equal Circles
	Curiosity 89. An Angle Property of the Miquel Point of a Triangle
	Curiosity 90. Yet More on the Miquel Relationship
	Curiosity 91. Three Intersecting Circles Generate Similar Triangles and More
	Curiosity 92. Equal Circles Generated by Three Equal Circles with a Common Point
	Curiosity 93. Congruent Figures Generated by Three Equal Circles with a Common Point
	Curiosity 94. Fun with Intersecting Circles
	Curiosity 95. Four Equal Circles Generate a Circumscribable Quadrilateral
	Curiosity 96. The Reuleaux Triangle
	Curiosity 97. Some Other Shapes with Constant Breadth
	Curiosity 98. Introducing the Radical Axis of Intersecting Circles
	Curiosity 99. The Radical Axis of Tangent Circles
	Curiosity 100. The Radical Axis of External Non-Intersecting Circles
	Curiosity 101. The Radical Axis of Internal Non-Intersecting Circles
	Curiosity 102. A Coincidence of Intersecting Circles
	Curiosity 103. The Radical Center of Three Circles
	Curiosity 104. A Surprising Result from a Randomly Selected Point Outside a Circle
	Curiosity 105. Unexpected Points on a Circle’s Tangent
	Curiosity 106. Points on a Secant Generate Points on a Special Circumcircle
	Curiosity 107. An External Circle Generates an Internal Circle
	Curiosity 108. Two Circles Generating Concyclic Points
	Curiosity 109. Generating Semicircles on the Sides of a Square
	Curiosity 110. Two Circles Generating an Unexpected Equilateral Triangle
	Curiosity 111. Two Equal Circles Produce an Isosceles Triangle
	Curiosity 112. Two Equal Circles Generate Another Isosceles Triangle
	Curiosity 113. An Unexpected Angle Bisector
	Curiosity 114. An Amazing Angle Equality in Intersecting Circles
	Curiosity 115. A Cyclic Quadrilateral Generates Two Circles with Perpendicular Radii
	Curiosity 116. Circles That Generate an Equality of Seemingly Unrelated Segments
	Curiosity 117. Related Equilateral Triangles Inscribed in Concentric Circles
	Curiosity 118. Intersecting Circles Surprisingly Generate Equal Lines
	Curiosity 119. Two Intersecting Circles Produce an Unexpected Line Segment Relationship
	Curiosity 120. A Quadrilateral with Two Intersecting Circles Generates Yet Another Circle
	Curiosity 121. The Broken Chord
	Curiosity 122. Tangents Yield Equal Segments
	Curiosity 123. The Fourth Corner of a Rectangle Lies on a Special Circle
	Curiosity 124. Creating Equal Chords in Two Circles
	Curiosity 125. An Unusual Angle Bisector Construction
	Curiosity 126. A Square Derived from Two Circles with a Surprising Area
	Curiosity 127. An Unexpected Circumcircle Center
	Curiosity 128. Chords of a Constant Length
	Curiosity 129. An Unexpected Equality of Two Special Chords
	Curiosity 130. A Surprising Common Point of Three Circles
	Curiosity 131. Perpendicular Circles Create a Diameter
Proofs of the Circle Curiosities
	Curiosity 1. A Triangle Defines a Circle
	Curiosity 2. A Circle Tangent is Perpendicular to the Radius in the Point of Tangency
	Curiosity 3. Circle Tangents from a Common Point are of Equal Length
	Curiosity 4. The Angle Between Circle Tangents from a Common Point is Supplementary to its Intercepted Arc
	Curiosity 5. Angle Between a Circle Tangent and a Chord
	Curiosity 6. The Angle Formed by Two Chords of a Circle
	Curiosity 7. Surprising Supplementary Angles
	Curiosity 8. The Angle Formed by a Tangent and a Secant
	Curiosity 9. The Angle Formed by Two Secants
	Curiosity 10. Trisecting an Angle Using a Circle
	Curiosity 11. A Rather Surprising Equality
	Curiosity 12. A Strange Appearance of Parallel Lines
	Curiosity 13. Unexpected Complementary Angles in a Circle
	Curiosity 14. Surprising Squares Equality in a Circle
	Curiosity 15. A Surprise of a Circle Presenting a Familiar Triangle
	Curiosity 16. A Surprising Perpendicularity
	Curiosity 17. Unexpected Concyclic Points
	Curiosity 18. Circle Chords Generate a Constant Angle Sum
	Curiosity 19. A Circle Helps Divide a Line Segment into Two Equal Parts
	Curiosity 20. The Famous Butterfly Relationship
	Curiosity 21. A Circle Unexpectedly Produces Parallel Lines
	Curiosity 22. The Unexpected Constant Point on a Circle
	Curiosity 23. Circle Tangents Generate Perpendicular Lines and Proportional Segments
	Curiosity 24. An Unexpected Circle Diameter
	Curiosity 25. An Unexpected Rectangle in a Circle
	Curiosity 26. Parallel Angle Bisectors
	Curiosity 27. Noteworthy Perpendiculars Generated by a Semicircle
	Curiosity 28. An Unexpected Angle Bisector in a Semicircle
	Curiosity 29. Tangent Circles on a Semicircle
	Curiosity 30. A Point on the Circumcircle of an Isosceles Triangle
	Curiosity 31. A Point on the Circumcircle of a Square
	Curiosity 32. A Point on the Circumcircle of a Regular Pentagon
	Curiosity 33. A Point on the Circumcircle of a Regular Hexagon
	Curiosity 34. The Tangent to the Circumscribed Circle of an Equilateral Triangle
	Curiosity 35. Surprising Concyclic Points
	Curiosity 36. The Midpoints of the Arcs of a Circle Produce Unexpected Parallels
	Curiosity 37. The Circumscribed Circle and the Altitudes of a Triangle Reveal an Unusual Equality
	Curiosity 38. The Unexpected Relationship Between the Midpoint of the Side of an Acute Triangle, its Circumcenter and its Orthocenter
	Curiosity 39. An Unexpected Concurrency with the Circumscribed Circle of the Triangle
	Curiosity 40. An Unexpected Angle Bisector Hidden Within the Circumscribed Circle
	Curiosity 41. Surprising Concyclic Points Generated by a Triangle and its Circumscribed Circle
	Curiosity 42. Unexpected Concyclic Points
	Curiosity 43. The Orthocenter and Two Vertices of a Triangle Generate a Circle Equal to the Circumcircle
	Curiosity 44. The Appearance of an Unexpected Isosceles Triangle
	Curiosity 45. Perplexing Parallels from Concyclic Points
	Curiosity 46. Tangents to the Circumcircle of a Triangle
	Curiosity 47. Tangents to the Nine-Point Circle
	Curiosity 48. An Adventure with the Inscribed Circle of a Triangle and a Special Circumcircle
	Curiosity 49. Adventures with Circumscribed and Inscribed Circles of a Triangle
	Curiosity 50. More Adventures with Circumscribed and Inscribed Circles of a Triangle
	Curiosity 51. How Four Centers of a Triangle can Generate an Unusual Cyclic Quadrilateral
	Curiosity 52. The Radii of an Inscribed and Escribed Circle Determine a Rectangle Area
	Curiosity 53. A Circumscribed Circle and an Inscribed Circle Generate Three Equal-Length Lines
	Curiosity 54. Three Circles with a Surprising Common Point
	Curiosity 55. The Famous Conway Circle
	Curiosity 56. A Surprising Collinearity with the Incenter of a Triangle — Sawayama’s Lemma
	Curiosity 57. The Surprising Relationship Between the Four Tangent Circles of a Triangle
	Curiosity 58. Two Tangent Circles Generate an Unexpected Right Angle
	Curiosity 59. The Unexpected Diameter of a Circle
	Curiosity 60. Doubling the Distance Between Centers
	Curiosity 61. Tangent Circles Generating a Circle Tangent to the Line of Centers
	Curiosity 62. An Unexpected Angle Bisector Generated by Externally Tangent Circles
	Curiosity 63. Unexpected Parallel Chords in Externally Tangent Circles
	Curiosity 64. Surprising Parallel Tangents of Externally Tangent Circles
	Curiosity 65. Two Externally Tangent Circles Generate a Line Half as Long as Their Line of Centers
	Curiosity 66. A Very Surprising Constant Tangent Length
	Curiosity 67. Two Circles Creating Parallel Lines Unexpectedly
	Curiosity 68. Two Circles Create an Angle Bisector
	Curiosity 69. Tangent Circles Produce Unusual Line Products
	Curiosity 70. Equal Angles Appear in Internally Tangent Circles
	Curiosity 71. Incredible Tangent Lengths in Tangent Circles
	Curiosity 72. Symmetric Equal Angles
	Curiosity 73. An Amazing Rolling Result
	Curiosity 74. Semicircles — The Arbelos
	Curiosity 75. An Arbelos Bisection
	Curiosity 76. More on the Common Arbelos Bisection Point
	Curiosity 77. Yet More on the Arbelos
	Curiosity 78. Another Arbelos Surprise
	Curiosity 79. Some Further Entertainment with the Arbelos
	Curiosity 80. A Surprising Area Equality Created by Semicircles
	Curiosity 81. Another Surprising Area Equality Created by Semicircles
	Curiosity 82. An Amazing Curve Division into Two Equal Parts
	Curiosity 83. An Unexpected Bisection
	Curiosity 84. A Circle-Related Point with a Double Function
	Curiosity 85. Three Circles Through Two Common Points Generate a Surprising Circle
	Curiosity 86. Intersecting Circles Create Five Concyclic Points
	Curiosity 87. Concyclic Centers of Three Related Circumcircles
	Curiosity 88. Two Unequal Circles Counterintuitively Produce Two Equal Circles
	Curiosity 89. An Angle Property of the Miquel Point of a Triangle
	Curiosity 90. Yet More on the Miquel Relationship
	Curiosity 91. Three Intersecting Circles Generate Similar Triangles and More
	Curiosity 92. Equal Circles Generated by Three Equal Circles with a Common Point
	Curiosity 93. Congruent Figures Generated by Three Equal Circles with a Common Point
	Curiosity 94. Fun with Intersecting Circles
	Curiosity 95. Four Equal Circles Generate a Circumscribable Quadrilateral
	Curiosity 96. The Reuleaux Triangle
	Curiosity 97. Some Other Shapes with Constant Breadth
	Curiosity 98. Introducing the Radical Axis of Intersecting Circles
	Curiosity 99. The Radical Axis of Tangent Circles
	Curiosity 100. The Radical Axis of External Non-Intersecting Circles
	Curiosity 101. The Radical Axis of Internal Non-Intersecting Circles
	Curiosity 102. A Coincidence of Intersecting Circles
	Curiosity 103. The Radical Center of Three Circles
	Curiosity 104. A Surprising Result from a Randomly Selected Point Outside a Circle
	Curiosity 105. Unexpected Points on a Circle’s Tangent
	Curiosity 106. Points on a Secant Generate Points on a Special Circumcircle
	Curiosity 107. An External Circle Generates an Internal Circle
	Curiosity 108. Two Circles Generating Concyclic Points
	Curiosity 109. Generating Semicircles on the Sides of a Square
	Curiosity 110. Two Circles Generating an Unexpected Equilateral Triangle
	Curiosity 111. Two Equal Circles Produce an Isosceles Triangle
	Curiosity 112. Two Equal Circles Generate Another Isosceles Triangle
	Curiosity 113. An Unexpected Angle Bisector
	Curiosity 114. An Amazing Angle Equality in Intersecting Circles
	Curiosity 115. A Cyclic Quadrilateral Generates Two Circles with Perpendicular Radii
	Curiosity 116. Circles That Generate an Equality of Seemingly Unrelated Segments
	Curiosity 117. Related Equilateral Triangles Inscribed in Concentric Circles
	Curiosity 118. Intersecting Circles Surprisingly Generate Equal Lines
	Curiosity 119. Two Intersecting Circles Produce an Unexpected Line Segment Relationship
	Curiosity 120. A Quadrilateral with Two Intersecting Circles Generates Yet Another Circle
	Curiosity 121. The Broken Chord
	Curiosity 122. Tangents Yield Equal Segments
	Curiosity 123. The Fourth Corner of a Rectangle Lies on a Special Circle
	Curiosity 124. Creating Equal Chords in Two Circles
	Curiosity 125. An Unusual Angle Bisector Construction
	Curiosity 126. A Square Derived from Two Circles with a Surprising Area
	Curiosity 127. An Unexpected Circumcircle Center
	Curiosity 128. Chords of a Constant Length
	Curiosity 129. An Unexpected Equality of Two Special Chords
	Curiosity 130. A Surprising Common Point of Three Circles
	Curiosity 131. Perpendicular Circles create a Diameter
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 and the Power of a Point (see Figure A5)
		A6: The Law of Sines and the Law of Cosines (see Figure A6)
	B: Less Familiar Tools – However, Useful and Fascinating
		B1: Ceva’s Theorem
		B2: Ceva’s Theorem Extended
		B3: Ptolemy’s Theorem
		B4: Isometries: Reflection and Rotation
		B5: Homothety and Similarity
Index