Geometric Gems: An Appreciation For Geometric Curiosities. Volume II: The Wonders Of Quadrilaterals

About the Authors
Acknowledgments
Introduction
Quadrilateral Curiosities
	Introducing the Quadrilateral
	Curiosity 1. Quadrilaterals in Triangles
	Curiosity 2. Comparing the Sum of Diagonals to the Sum of Opposite Sides of a Quadrilateral
	Curiosity 3. Partitioning a Trapezoid into Two Equal Areas
	Curiosity 4. The Unexpected Triangle that is Half the Area of a Trapezoid
	Curiosity 5. The Surprising Line Joining the Midpoints of the Diagonals of a Trapezoid
	Curiosity 6. An Unexpected Collinearity in a Trapezoid
	Curiosity 7. Surprising Concurrency Generated by a Trapezoid
	Curiosity 8. Another Surprising Concurrency Generated by a Trapezoid
	Curiosity 9. A Curious Angle Equality
	Curiosity 10. A Curious Line Segment Equality
	Curiosity 11. A Surprising Partitioning of a General Quadrilateral
	Curiosity 12. Trisected Sides of a Quadrilateral Determine One-Third of a Quadrilateral
	Curiosity 13. Partitioning a Quadrilateral in Half
	Curiosity 14. Partitioning a Quadrilateral into Three Equal Area Regions
	Curiosity 15. Another Partition of a Quadrilateral into Three Equal Area Regions
	Curiosity 16. When Two Quadrilaterals Share Midpoints
	Curiosity 17. Locating the Centroid of a Quadrilateral
	Curiosity 18. Centroids Can Generate a Similar Quadrilateral in the Size Ratio 1:9
	Curiosity 19. The Minimum Distance Point of a Quadrilateral
	Curiosity 20. A Rhombus Curiosity
	Curiosity 21. A Counterintuitive Comparison of Parallelograms
	Curiosity 22. Proportionality of Parallelograms
	Curiosity 23. An Unexpected Concurrency in Partner Parallelograms
	Curiosity 24. Partitioning a Parallelogram in Half
	Curiosity 25. Partitioning a Trapezoid in Half
	Curiosity 26. Partitioning a Parallelogram into Thirds
	Curiosity 27. Trisecting a Diagonal of a Parallelogram
	Curiosity 28. Another Clever Trisection of a Diagonal of a Parallelogram
	Curiosity 29. Yet Another Trisection Point on a Diagonal of a Parallelogram
	Curiosity 30. Relating Parallelogram Side Lengths to Diagonal Lengths
	Curiosity 31. An Unexpected Perpendicularity in a Parallelogram
	Curiosity 32. Curious Surprises Embedded in a Parallelogram
	Curiosity 33. Triangles Generating Parallelograms without Constructing Parallel Lines
	Curiosity 34. Creating a Parallelogram with One-Ninth the Area of the Original Parallelogram
	Curiosity 35. How Bisecting and Trisecting Angles of a Parallelogram Creates Another Parallelogram
	Curiosity 36. Using a Parallelogram to Create a Square
	Curiosity 37. Parallelogram Generates an Equilateral Triangle
	Curiosity 38. Exposing Secret Products Along the Sides of a Parallelogram
	Curiosity 39. Unusual Equal Areas Created in a Parallelogram
	Curiosity 40. Equal Triangle Areas Simply Generated by a Parallelogram
	Curiosity 41. More Equal Triangle Areas Hidden in a Parallelogram
	Curiosity 42. Another Example of Equal Areas in a Parallelogram
	Curiosity 43. Unexpected Perpendicularity Emerging from a Parallelogram
	Curiosity 44. The Varignon Parallelogram and Its Special Properties
	Curiosity 45. A Truly Spectacular Parallelogram
	Curiosity 46. More on the Centerpoint of a Quadrilateral
	Curiosity 47. More Surprises in the Parallelogram
	Curiosity 48. Special Quadrilaterals Yield Varignon Rectangles
	Curiosity 49. Very Special Quadrilaterals Yield Varignon Squares
	Curiosity 50. Other Special Quadrilaterals Can Yield a Varignon Rhombus
	Curiosity 51. The Wittenbauer Parallelogram
	Curiosity 52. A Surprising Equality in a Rectangle
	Curiosity 53. An Unexpected Constant of Points on a Rectangle
	Curiosity 54. An Unexpected Perpendicularity in a Rectangle
	Curiosity 55. A Powerful Point in a Rectangle that Generates Equal Sums of Squares
	Curiosity 56. Equating Areas of a Square and a Rectangle Attached to a Larger Square
	Curiosity 57. The Placement of Squares and Rhombuses
	Curiosity 58. Connected Squares Generate Another Square
	Curiosity 59. An Unusual Arrangement Can Lead to Unexpected Concyclic Points
	Curiosity 60. Unexpected Concyclic Points in a Rectangle
	Curiosity 61. A Square on a Right Triangle
	Curiosity 62. Perpendicularity in a Square and an Unexpected Equality
	Curiosity 63. A Surprising Concurrency in a Square
	Curiosity 64. A Surprising Equality in a Square
	Curiosity 65. An Astounding Equality in a Square
	Curiosity 66. Startling Area Relationships in a Special Partitioning of a Square
	Curiosity 67. An Equality Arises When a Square and Rectangle Share a Common Diagonal
	Curiosity 68. Surprising Area Equality Within a Square
	Curiosity 69. Astounding Equal-Area Triangles Produced by a Square
	Curiosity 70. More Equal Areas Generated by an Initial Square
	Curiosity 71. Two Squares Produce an Unforeseen Equality
	Curiosity 72. Surprising Result of the Angle Bisectors of a Complete Quadrilateral
	Curiosity 73. Perpendiculars from the Vertices of a Quadrilateral to a Randomly Drawn Line
	Curiosity 74. More Relationships in Quadrilaterals
	Curiosity 75. Two Equal Adjacent Sides of a Quadrilateral Produce Equal Angles
	Curiosity 76. Two Equal Opposite Sides of a Quadrilateral Produce Equal Angles
	Curiosity 77. Angle Formed by Bisectors of Consecutive Angles of a Quadrilateral
	Curiosity 78. A Quadrilateral with Two Consecutive Right Angles Has Perpendicular Angle Bisectors
	Curiosity 79. The Comparative Areas of the Square on the Quadrilateral
	Curiosity 80. Squares on Quadrilateral Sides Produce Equal-Area Triangles
	Curiosity 81. Equality of Pairs of Squares on Diagonals and Midlines
	Curiosity 82. Introducing the Cyclic Quadrilateral
	Curiosity 83. Another Wonderful Property of the Cyclic Quadrilateral
	Curiosity 84. An Astounding Angle Property of the Cyclic Quadrilateral
	Curiosity 85. Generating Cyclic Quadrilaterals
	Curiosity 86. The Orthocenter of a Triangle Generates Lots of Cyclic Quadrilaterals
	Curiosity 87. More on a Cyclic Quadrilateral
	Curiosity 88. Angle Bisectors of a Quadrilateral Create a Cyclic Quadrilateral
	Curiosity 89. Angle Bisectors of a Parallelogram Create a Rectangle
	Curiosity 90. Angle Bisectors of a Rectangle Create a Square
	Curiosity 91. Unexpected Concyclic Points Emanating from an Isosceles Trapezoid
	Curiosity 92. An Unexpected Perpendicularity Within a Cyclic Quadrilateral
	Curiosity 93. Unusual Lines Bisecting Each Other in a Cyclic Quadrilateral
	Curiosity 94. More Unusual Lines Bisecting Each Other in a Cyclic Quadrilateral
	Curiosity 95. Partitioning a Quadrilateral into Two Equal Areas
	Curiosity 96. Another Surprise in a Cyclic Quadrilateral
	Curiosity 97. An Interesting Orthocenter Property in a Parallelogram
	Curiosity 98. An Unexpected Cyclic Quadrilateral
	Curiosity 99. The Incenters of the Four Triangles of a Cyclic Quadrilateral Determine a Rectangle
	Curiosity 100. A Cyclic Quadrilateral Generates a Rhombus
	Curiosity 101. A Cyclic Quadrilateral Generates a Parallelogram
	Curiosity 102. A Cyclic Quadrilateral Surprise
	Curiosity 103. A Cyclic Quadrilateral Generated by Another Cyclic Quadrilateral
	Curiosity 104. Perpendiculars to a Diagonal and a Side from a Vertex of a Special Cyclic Quadrilateral Determine the Midpoint of the Other Diagonal
	Curiosity 105. Perpendiculars from Midpoints of a Cyclic Quadrilateral are Concurrent
	Curiosity 106. Perpendicular Surprises in a Cyclic Quadrilateral
	Curiosity 107. A Cyclic Quadrilateral Produces another Surprise Perpendicularity
	Curiosity 108. Unexpected Parallels from Perpendiculars to Cyclic Quadrilateral Sides
	Curiosity 109. Surprises in Cyclic Quadrilaterals with Perpendicular Diagonals
	Curiosity 110. Yet Another Surprise Cyclic Quadrilateral Perpendicularity
	Curiosity 111. A Cyclic Quadrilateral with Perpendicular Diagonals Produces a Parallelogram
	Curiosity 112. Another Unanticipated Property of Cyclic Quadrilaterals with Perpendicular Diagonals
	Curiosity 113. A Cyclic Quadrilateral with Perpendicular Diagonals Produces Concyclic Points
	Curiosity 114. Another Unanticipated Property of Cyclic Quadrilaterals with Perpendicular Diagonals
	Curiosity 115. A Cyclic Quadrilateral with Perpendicular Diagonals Produces Equal-Area Triangles
	Curiosity 116. A Counterintuitive Property of Quadrilaterals with Perpendicular Diagonals
	Curiosity 117. Equal Sums of Areas of Squares on Opposite Sides of a Particular Quadrilateral
	Curiosity 118. An Unexpected Perpendicularity Arises in a Cyclic Quadrilateral
	Curiosity 119. Relating Cyclic Quadrilateral Diagonals and Sides
	Curiosity 120. The Famous Ptolemy’s Theorem
	Curiosity 121. A Curious Parallelogram Relationship
	Curiosity 122. The Equilateral Triangle in a Cyclic Quadrilateral
	Curiosity 123. Angle Bisectors in Cyclic Quadrilaterals
	Curiosity 124. A Surprise Angle Bisector in a Special Cyclic Quadrilateral
	Curiosity 125. Perpendiculars to the Cyclic Quadrilateral Sides from a Point on its Circumcircle
	Curiosity 126. Amazements with a Cyclic Quadrilateral
	Curiosity 127. Brahmagupta’s Famous Theorem
	Curiosity 128. Strange Equality of Squares
	Curiosity 129. Proportionality Generates Equality of Areas
	Curiosity 130. Partner Squares on Quadrilateral Sides Generates an Equality
	Curiosity 131. The Line Joining the Midpoints of the Diagonals of a Quadrilateral – Newton Line
	Curiosity 132. Partitioning a Quadrilateral into Two Pairs of Triangles with Equal Area Sums
	Curiosity 133. The Circumscribable Quadrilateral
	Curiosity 134. The Newton Line in a Circumscribable Quadrilateral
	Curiosity 135. More About the Newton Line in a Quadrilateral with an Inscribed Circle
	Curiosity 136. Determining a Triangle One-Quarter of the Area of a Related Quadrilateral
	Curiosity 137. The Midpoints of the Diagonals of a Complete Quadrilateral
	Curiosity 138. Partitioning a Quadrilateral into Four Equal Regions
	Curiosity 139. Partitioning a Quadrilateral into Two Pairs of Equal-Area Triangles
	Curiosity 140. A Circle with an Inscribed and Circumscribed Quadrilateral Produces Collinear Points
Proofs of Quadrilateral Curiosities
	Curiosity 1. Quadrilaterals in Triangles
	Curiosity 2. Comparing the Sum of Diagonals to the Sum of Opposite Sides of a Quadrilateral
	Curiosity 3. Partitioning a Trapezoid into Two Equal Areas
	Curiosity 4. The Unexpected Triangle that is Half the Area of a Trapezoid
	Curiosity 5. The Surprising Line Joining the Midpoints of the Diagonals of a Trapezoid
	Curiosity 6. An Unexpected Collinearity in a Trapezoid
	Curiosity 7. Surprising Concurrency Generated by a Trapezoid
	Curiosity 8. Another Surprising Concurrency Generated by a Trapezoid
	Curiosity 9. A Curious Angle Equality
	Curiosity 10. A Curious Line Segment Equality
	Curiosity 11. A Surprising Partitioning of a General Quadrilateral
	Curiosity 12. Trisected Sides of a Quadrilateral Determine One-Third of a Quadrilateral
	Curiosity 13. Partitioning a Quadrilateral in Half
	Curiosity 14. Partitioning a Quadrilateral into Three Equal Area Regions
	Curiosity 15. Another Partitioning of a Quadrilateral into Three Equal Area Regions
	Curiosity 16. When Two Quadrilaterals Share Midpoints
	Curiosity 17. Locating the Centroid of a Quadrilateral
	Curiosity 18. Centroids Can Generate a Similar Quadrilateral in the Size-Ratio 1:9
	Curiosity 19. The Minimum Distance Point of a Quadrilateral
	Curiosity 20. A Rhombus Curiosity
	Curiosity 21. A Counterintuitive Comparison of Parallelograms
	Curiosity 22. Proportionality of Parallelograms
	Curiosity 23. An Unexpected Concurrency in Partner Parallelograms
	Curiosity 24. Partitioning a Parallelogram in Half
	Curiosity 25. Partitioning a Trapezoid in Half
	Curiosity 26. Partitioning a Parallelogram into Thirds
	Curiosity 27. Trisecting a Diagonal of a Parallelogram
	Curiosity 28. Another Clever Trisection of a Diagonal of a Parallelogram
	Curiosity 29. Yet Another Trisection Point on a Diagonal of a Parallelogram
	Curiosity 30. Relating Parallelogram Side Lengths to Diagonal Lengths
	Curiosity 31. An Unexpected Perpendicularity in a Parallelogram
	Curiosity 32. Curious Surprises Embedded in a Parallelogram
	Curiosity 33. Triangles Generating Parallelograms without Constructing Parallel Lines
	Curiosity 34. Creating a Parallelogram with One-Ninth the Area of the Original Parallelogram
	Curiosity 35. How Bisecting and Trisecting Angles of a Parallelogram Creates Another Parallelogram
	Curiosity 36. Using a Parallelogram to Create a Square
	Curiosity 37. Parallelogram Generates an Equilateral Triangle
	Curiosity 38. Exposing Secret Products Along the Sides of a Parallelogram
	Curiosity 39. Unusual Equal Areas Created in a Parallelogram
	Curiosity 40. Equal Triangle Areas Simply Generated by a Parallelogram
	Curiosity 41. More Equal Triangle Areas Hidden in a Parallelogram
	Curiosity 42. Another Example of Equal Areas in a Parallelogram
	Curiosity 43. Unexpected Perpendicularity Emerging from a Parallelogram
	Curiosity 44. The Varignon Parallelogram and Its Special Properties
	Curiosity 45. A Truly Spectacular Parallelogram
	Curiosity 46. More on the Centerpoint of a Quadrilateral
	Curiosity 47. More Surprises in the Parallelogram
	Curiosity 48. Special Quadrilaterals Yield Varignon Rectangles
	Curiosity 49. Very Special Quadrilaterals Yield Varignon Squares
	Curiosity 50. Other Special Quadrilaterals Can Yield a Varignon Rhombus
	Curiosity 51. The Wittenbauer Parallelogram
	Curiosity 52. A Surprising Equality in a Rectangle
	Curiosity 53. An Unexpected Constant of Points on a Rectangle
	Curiosity 54. An Unexpected Perpendicularity in a Rectangle
	Curiosity 55. A Powerful Point in a Rectangle that Generates Equal Sums of Squares
	Curiosity 56. Equating Areas of a Square and a Rectangle Attached to a Larger Square
	Curiosity 57. The Placement of Squares and Rhombuses
	Curiosity 58. Connected Squares Generate Another Square
	Curiosity 59. An Unusual Arrangement Can Lead to Unexpected Concyclic Points
	Curiosity 60. Unexpected Concyclic Points in a Rectangle
	Curiosity 61. A Square on a Right Triangle
	Curiosity 62. Perpendicularity in a Square and an Unexpected Equality
	Curiosity 63 A Surprising Concurrency in a Square
	Curiosity 64. A Surprising Equality in a Square
	Curiosity 65. An Astounding Equality in a Square
	Curiosity 66. Startling Area Relationships in a Special Partitioning of a Square
	Curiosity 67. An Equality Arises When a Square and Rectangle Share a Common Diagonal
	Curiosity 68. Surprising Area Equality Within a Square
	Curiosity 69. Astounding Equal-Area Triangles Produced by a Square
	Curiosity 70. More Equal Areas Generated by an Initial Square
	Curiosity 71. Two Squares Produce an Unforeseen Equality
	Curiosity 72. Surprising Result of the Angle Bisectors of a Complete Quadrilateral
	Curiosity 73. Perpendiculars from the Vertices of a Quadrilateral to a Randomly Drawn Line
	Curiosity 74. More Relationships in Quadrilaterals
	Curiosity 75. Two Equal Adjacent Sides of a Quadrilateral Produce Equal Angles
	Curiosity 76. Two Equal Opposite Sides of a Quadrilateral Produce Equal Angles
	Curiosity 77. Angle Formed by Bisectors of Consecutive Angles of a Quadrilateral
	Curiosity 78. A Quadrilateral with Two Consecutive Right Angles Has Perpendicular Angle Bisectors
	Curiosity 79. The Comparative Areas of the Square on the Quadrilateral
	Curiosity 80. Squares on Quadrilateral Sides Produce Equal-Area Triangles
	Curiosity 81. Equality of Pairs of Squares on Diagonals and Midlines
	Curiosity 82. Introducing the Cyclic Quadrilateral
	Curiosity 83. Another Wonderful Property of the Cyclic Quadrilateral
	Curiosity 84. An Astounding Angle Property of the Cyclic Quadrilateral
	Curiosity 85. Generating Cyclic Quadrilaterals
	Curiosity 86. The Orthocenter of a Triangle Generates Lots of Cyclic Quadrilaterals
	Curiosity 87. More on a Cyclic Quadrilateral
	Curiosity 88. Angle Bisectors of a Quadrilateral Create a Cyclic Quadrilateral
	Curiosity 89. Angle Bisectors of a Parallelogram Create a Rectangle
	Curiosity 90. Angle Bisectors of a Rectangle Create a Square
	Curiosity 91. Unexpected Concyclic Points Emanating from an Isosceles Trapezoid
	Curiosity 92. An Unexpected Perpendicularity Within a Cyclic Quadrilateral
	Curiosity 93. Unusual Lines Bisecting Each Other in a Cyclic Quadrilateral
	Curiosity 94. More Unusual Lines Bisecting Each Other in a Cyclic Quadrilateral
	Curiosity 95. Partitioning a Quadrilateral into Two Equal Areas
	Curiosity 96. Another Surprise in a Cyclic Quadrilateral
	Curiosity 97. An Interesting Orthocenter Property in a Parallelogram
	Curiosity 98. An Unexpected Cyclic Quadrilateral
	Curiosity 99. The Incenters of the Four Triangles of a Cyclic Quadrilateral Determine a Rectangle
	Curiosity 100. A Cyclic Quadrilateral Generates a Rhombus
	Curiosity 101. A Cyclic Quadrilateral Generates a Parallelogram
	Curiosity 102. A Cyclic Quadrilateral Surprise
	Curiosity 103. A Cyclic Quadrilateral Generated by Another Cyclic Quadrilateral
	Curiosity 104. Perpendiculars to a Diagonal and a Side from a Vertex of a Special Cyclic Quadrilateral Determine the Midpoint of the Other Diagonal
	Curiosity 105. Perpendiculars from Midpoints of a Cyclic Quadrilateral are Concurrent
	Curiosity 106. Perpendicular Surprises in a Cyclic Quadrilateral
	Curiosity 107. A Cyclic Quadrilateral Produces Another Surprise Perpendicularity
	Curiosity 108. Unexpected Parallels from Perpendiculars to Cyclic Quadrilateral Sides.
	Curiosity 109. Surprises with Cyclic Quadrilaterals with Perpendicular Diagonals
	Curiosity 110. Yet Another Surprise Cyclic Quadrilateral Perpendicularity
	Curiosity 111. A Cyclic Quadrilateral with Perpendicular Diagonals Produces a Parallelogram
	Curiosity 112. Another Unanticipated Property of Cyclic Quadrilaterals with Perpendicular Diagonals
	Curiosity 113. A Cyclic Quadrilateral with Perpendicular Diagonals Produces Concyclic Points
	Curiosity 114. Another Unanticipated Property of Cyclic Quadrilaterals with Perpendicular Diagonals
	Curiosity 115. A Cyclic Quadrilateral with Perpendicular Diagonals Produces Equal-Area Triangles
	Curiosity 116. A Counterintuitive Property of Quadrilaterals with Perpendicular Diagonals
	Curiosity 117. Equal Sums of Areas of Squares on Opposite Sides of a Particular Quadrilateral
	Curiosity 118. An Unexpected Perpendicularity Arises in a Cyclic Quadrilateral
	Curiosity 119. Relating Cyclic Quadrilateral Diagonals and Sides
	Curiosity 120. The Famous Ptolemy’s Theorem
	Curiosity 121. A Curious Parallelogram relationship
	Curiosity 122. The Equilateral Triangle in a Cyclic Quadrilateral
	Curiosity 123. Angle Bisectors in Cyclic Quadrilaterals
	Curiosity 124. A Surprise Angle Bisector in a Special Cyclic Quadrilateral
	Curiosity 125. Perpendiculars to the Cyclic Quadrilateral Sides from a Point on its Circumcircle
	Curiosity 126. Amazements with a Cyclic Quadrilateral
	Curiosity 127. Brahmagupta’s Famous Theorem
	Curiosity 128. Strange Equality of Squares
	Curiosity 129. Proportionality Generates Equality of Areas
	Curiosity 130. Partner Squares on Quadrilateral Sides Generates an Equality
	Curiosity 131. The Line Joining the Midpoints of the Diagonals of a Quadrilateral – Newton Line
	Curiosity 132. Partitioning a Quadrilateral into Two Pairs of Triangles with Equal Area Sums
	Curiosity 133. The Circumscribable Quadrilateral
	Curiosity 134. The Newton Line in a Circumscribable Quadrilateral
	Curiosity 135. More About the Newton Line in a Quadrilateral with an Inscribed Circle
	Curiosity 136. Determining a Triangle One-Quarter of the Area of a Related Quadrilateral
	Curiosity 137. The Midpoints of the Diagonals of a Complete Quadrilateral
	Curiosity 138. Partitioning a Quadrilateral into Four Equal Regions
	Curiosity 139. Partitioning a Quadrilateral into Two Pairs of Equal-Area Triangles
	Curiosity 140. A Circle with an Inscribed and Circumscribed Quadrilateral Produces Collinear Points
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
		A4: Angles Related to a Circle
		A5: Tangents, Secants, and Chords: Segments of a Circle and the Power of a Point
		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: Ceva’s Theorem Extended
		B6: Desargues’s Theorem
		B7: Isometries: Reflection and Rotation
		B8: Homothety and Similarity
		B9: Simson’s Theorem
		B10: Heron’s Formula
Index
Contents