Geometry Proof Challenges

Tim Brzezinski, Apr 25, 2017

Table of Contents

  1. Lines and Angles
    1. Cut-The-Knot-Action (5)!
    2. Animation 214
    3. Cut-the-Knot-Action (3)!
    4. Animation 188
    5. GoGeometry Action 40!
    6. Animation 259
    7. GoGeometry Action 41!
    8. Animation 260
    9. GoGeometry Action 58!
    10. Animation 278
    11. GoGeometry Action 63!
    12. GoGeometry Action 65!
    13. Animation 285
    14. Concave Quadrilateral Craziness! (GoGeometry Action 80)
    15. Hexagonal Surprise! (GoGeometry Action 85)
    16. GoGeometry Action 123!
    17. GoGeometry Action 143!
    18. SolveMyMaths Action 2!
    19. GoGeometry Action 158!
    20. GoGeometry Action 159!
    21. GoGeometry Action 168!
    22. GoGeometry Action 169!
    23. SolveMyMaths Action 7!
    24. Regular Polygon Phenomena!
    25. Regular Nonagon Simplicity (GoGeometry Action 175!)
    26. More Nonagon Wonders (GoGeometry Action 176!)
  2. Triangles
    1. Medians & Equal Areas!
    2. Animation 74
    3. Animation 90
    4. Euler Line (Informal Investigation)
    5. Euler Line Generator
    6. Morley Action!
    7. Animation 113
    8. Napoleon Motion!
    9. Animation 14
    10. Hexagonal Napoleon Theorem?
    11. Animation 172
    12. Viviani Action! (REVAMPED)
    13. Equilateral Action (1)!
    14. Animation 91
    15. Equilateral Action (2)!
    16. Animation 94
    17. Equilateral Action (3)!
    18. Animation 107
    19. Equilateral Action (4)!
    20. Animation 108
    21. Equilateral Triangle and Rectangle (Area Problem)
    22. Animation 154
    23. Equal Triangular Areas in a Regular Pentagon
    24. Animation 194
    25. NCTM Calendar Problem (11-2-2016)
    26. Animation 139
    27. 9 Point Circle Action
    28. Animation 34
    29. 9-Point Circle Action (Part 2)
    30. Animation 106
    31. 9-Point Circle Action (Part 3A)
    32. 9-Point Circle (Informal Investigation)
    33. Animation 71
    34. 9-Point Circle Action (Part 3B)
    35. 9-Point Circle Action (Part 4)
    36. Animation 53
    37. 9-Point Circle Action (Part 5)
    38. Animation 115
    39. Isosceles Triangle: Surprising Theorem!
    40. Animation 121
    41. Cut-The-Knot Action !!!
    42. More Triangle Action!!!
    43. Animation 163
    44. Triangle Median & Circles Action!!!
    45. Animation 164
    46. Desargues Action!!!
    47. Animation 209
    48. GoGeometry Action 10!
    49. Animation 219
    50. GoGeometry Action 14!
    51. Animation 223
    52. GoGeometry Action 17!
    53. Animation 228
    54. GoGeometry Action 18!
    55. Animation 229
    56. Cut-The-Knot-Action 8!
    57. Animation 230
    58. GoGeometry Action 20!
    59. Animation 233
    60. Cut-The-Knot Action 9!
    61. Animation 234
    62. Carnot Action! = GoGeometry Action 27!
    63. Animation 241
    64. GoGeometry Action 29!
    65. Animation 244
    66. GoGeometry Action 30!
    67. Animation 245
    68. Incircle + Excircle = GoGeometry Action 31!
    69. Animation 246
    70. GoGeometry Action 32!
    71. Animation 247
    72. Cut-The-Knot Action 10!
    73. Animation 248
    74. GoGeometry Action 34!
    75. Animation 250
    76. Cut-The-Knot-Action 12!
    77. Animation 251
    78. GoGeometry Action 35!
    79. Animation 252
    80. GoGeometry Action 37!
    81. Animation 254
    82. GoGeometry Action 38!
    83. Animation 255
    84. GoGeometry Action 39!
    85. Animation 258
    86. GoGeometry Action 43!
    87. Animation 263
    88. GoGeometry Action 44!
    89. Animation 264
    90. GoGeometry Action 51!
    91. Animation 270
    92. GoGeometry Action 53!
    93. Animation 272
    94. Euler Line Surprise! = GoGeometry Action 54
    95. Animation 273!
    96. GoGeometry Action 60!
    97. Animation 280
    98. Animation 283
    99. GoGeometry Action 64!
    100. Animation 284
    101. GoGeometry Action 66!
    102. Animation 286
    103. GoGeometry Action 68!
    104. Animation 288
    105. GoGeometry Action 74!
    106. GoGeometry Action 75!
    107. GoGeometry Action 76!
    108. GoGeometry Action 78!
    109. GoGeometry Action 82!
    110. GoGeometry Action 83!
    111. Cut-the-Knot Action 15!
    112. Equilateral Triangle + Semicircle = ? (GoGeometry Action 88)
    113. GoGeometry Action 89!
    114. GoGeometry Action 90!
    115. GoGeometry Action 91!
    116. GoGeometry Action 93!
    117. GoGeometry Action 97!
    118. GoGeometry Action 101 !
    119. GoGeometry Action 103!
    120. GoGeometry Action 104!
    121. GoGeometry Action 105!
    122. GoGeometry Action 106!
    123. GoGeometry Action 107!
    124. GoGeometry Action 109!
    125. SolveMyMaths Action 1!
    126. GoGeometry Action 110!
    127. GoGeometry Action 112!
    128. GoGeometry Action 113!
    129. GoGeometry Action 115!
    130. GoGeometry Action 117!
    131. Blanchet Action = GoGeometry Action 118!
    132. GoGeometry Action 120!
    133. GoGeometry Action 121!
    134. GoGeometry Action 124!
    135. GoGeometry Action 125!
    136. Cut-The-Knot Action 18!
    137. Cut-The-Knot Action 19!
    138. GoGeometry Action 127!
    139. Incircle Extension = GoGeometry Action 132!
    140. Incircle Extension 2 = GoGeometry Action 133!
    141. GoGeometry Action 134!
    142. GoGeometry Action 135!
    143. GoGeometry Action 138!
    144. GoGeometry Action 147!
    145. GoGeometry Action 152!
    146. GoGeometry Action 156!
    147. GoGeometry Action 160!
    148. Regular Decagon Magic!
    149. GoGeometry Action 161!
    150. GoGeometry Action 162!
    151. GoGeometry Action 163!
    152. Equilateral Triangles Problem
    153. GoGeometry Action 166!
    154. Feuerbach Action! (GoGeometry Action 167)
    155. GoGeometry Action 170!
    156. GoGeometry Action 171!
    157. GoGeometry Action 174!
    158. GoGeometry Action 182!
    159. GoGeometry Action 183!
    160. GoGeometry Action 184!
    161. GoGeometry Action 185!
    162. GoGeometry Action 188!
    163. GoGeometry Action 189!
    164. Regular Octagon Phenomenon (GoGeometry Action 190)
  3. Parallelograms
    1. Finsler-Hadwiger Action!!!
    2. Animation 110
    3. Thebault Action!!!
    4. Animation 83
    5. Van Aubel Action!!!
    6. Animation 109
    7. Square Problem
    8. 3 Squares Problem
    9. Animation 120
    10. Harmonic Mean In an Isosceles Trapezoid
    11. Square Action! (Proof Problem 1)
    12. Cut-The-Knot Action (4)!
    13. Animation 213
    14. Rhombus Action + Sequel = GoGeometry Action 7!
    15. Animation 215
    16. GoGeometry Action 11!
    17. Animation 220
    18. GoGeometry Action 12!
    19. Animation 221
    20. GoGeometry Action 18!
    21. Animation 231
    22. GoGeometry Action 21!
    23. Animation 235
    24. GoGeometry Action 22!
    25. Animation 236
    26. GoGeometry Action 23!
    27. Animation 237
    28. GoGeometry Action 24!
    29. Animation 238
    30. GoGeometry Action 36!
    31. Animation 253
    32. Cut-the-Knot-Action 13!
    33. Animation 257
    34. GoGeometry Action 42!
    35. Animation 261
    36. Cut-The-Knot-Action 14!
    37. Animation 262
    38. GoGeometry Action 45!
    39. Animation 265
    40. GoGeometry Action 46!
    41. Animation 266
    42. GoGeometry Action 47!
    43. Animation 267
    44. Rhombus Surprise! (GoGeometry Action 57)
    45. Animation 277
    46. GoGeometry Action 67!
    47. Animation 287
    48. GoGeometry Action 69!
    49. Square + Rhombus = Constant Surprise! (GoGeometry Action 71)
    50. GoGeometry Action 73!
    51. Parallelogram? Why So? (GoGeometry Action 77)
    52. Square and Quarter Circle Action = GoGeometry Action 81!
    53. 3 Squares Problem (II)
    54. GoGeometry Action 94!
    55. GoGeometry Action 95!
    56. GoGeometry Action 96!
    57. GoGeometry Action 98!
    58. GoGeometry Action 99!
    59. GoGeometry Action 102 !
    60. Square + Regular Hexagon Action!
    61. GoGeometry Action 108!
    62. GoGeometry Action 114!
    63. GoGeometry Action 119!
    64. Cut-The-Knot Action 17!
    65. GoGeometry Action 126!
    66. GoGeometry Action 130!
    67. GoGeometry Action 136!
    68. GoGeometry Action 139!
    69. GoGeometry Action 140!
    70. Parallelogram + 2 Equilateral Triangles: How to Prove?
    71. GoGeometry Action 153!
    72. GoGeometry Action 165!
    73. Shearer Action 1! (Regular Octagon Problem)
    74. Shearer Action 6!
    75. GoGeometry Action 181!
    76. GoGeometry Action 186!
    77. GoGeometry Action 187!
  4. Geometric Constructions
    1. Bizzare Trisection?
    2. Animation 198
    3. Bizarre Trisection (V2) = GoGeometry Action 145!
    4. Angle Trisection by Paper Folding!
    5. Bizarre Trisection (V3) = GoGeometry Action 146!
  5. Circle Theorems
    1. Butterfly Theorem Action!
    2. Animation 132
    3. Not Your Everyday Chord & Tangent Theorem
    4. Animation 133
    5. Cut-The-Knot Action (2) !!!
    6. Animation 152
    7. Int+Ext Tangent Action!
    8. Animation 155
    9. Square, Tangent, and Two Congruent & Tangent Circles
    10. Animation 153
    11. GoGeometry Action 1!
    12. Animation 205
    13. GoGeometry Action 2!
    14. Animation 206
    15. GoGeometry Action 3!
    16. Animation 207
    17. GoGeometry Action 4!
    18. Animation 208
    19. GoGeometry Action 5!
    20. Animation 210
    21. GoGeometry Action 6!
    22. Animation 212
    23. GoGeometry Action 8!
    24. Animation 216
    25. Cut-The-Knot Action (6)!
    26. Animation 217
    27. GoGeometry Action 9!
    28. Animation 218
    29. GoGeometry Action 13!
    30. Animation 222
    31. GoGeometry Action 15!
    32. Animation 224
    33. GoGeometry Action 16!
    34. Animation 226
    35. Cut-The-Knot Action 7!
    36. Animation 227
    37. GoGeometry Action 19!
    38. Animation 232
    39. Thales Action + Sequel = GoGeometry Action 25!
    40. Animation 239
    41. GoGeometry Action 26!
    42. Animation 240
    43. Carnot Action! = GoGeometry Action 27!
    44. Animation 241
    45. Incircle + Excircle = GoGeometry Action 31!
    46. Animation 246
    47. Cut-The-Knot Action 10!
    48. Animation 248
    49. Cut-The-Knot-Action 11!
    50. Animation 249
    51. GoGeometry Action 48!
    52. Animation 268
    53. GoGeometry Action 49!
    54. GoGeometry Action 50!
    55. Animation 269
    56. GoGeometry Action 52!
    57. Animation 271
    58. 60-Degree Sector Surprise! (GoGeometry Action 55)
    59. Animation 274
    60. Neusis Logic Action = GoGeometry Action 56!
    61. Animation 275
    62. Animation 276
    63. GoGeometry Action 59!
    64. Animation 279
    65. GoGeometry Action 61!
    66. Animation 281
    67. GoGeometry Action 62!
    68. Animation 282
    69. GoGeometry Action 70!
    70. GoGeometry Action 72!
    71. GoGeometry Action 79!
    72. Geometric Mean Madness! (GoGeometry Action 84)
    73. Mickey Mouse Theorem! (Cut-the-Knot Action 16)
    74. More Geometric Mean Action! (GoGeometry Action 86)
    75. GoGeometry Action 87!
    76. GoGeometry Action 92!
    77. A Stronger Conclusion!
    78. GoGeometry Action 100!
    79. GoGeometry Action 116!
    80. GoGeometry Action 122!
    81. GoGeometry Action 128!
    82. GoGeometry Action 129!
    83. GoGeometry Action 131!
    84. GoGeometry Action 137!
    85. GoGeometry Action 142!
    86. Circle Optimization Challenge!
    87. SolveMyMaths Action 5!
    88. GoGeometry Action 148!
    89. Cyclic Quadrilateral: Quick Warm Up (GoGeometry Action 149!)
    90. Archimedes Insight = GoGeometry Action 150!
    91. More Archimedes Insight = GoGeometry Action 151!
    92. GoGeometry Action 154!
    93. GoGeometry Action 155!
    94. Archimedes Insight 3: GoGeometry Action 157!
    95. GoGeometry Action 164!
    96. GoGeometry Action 172!
    97. Shearer Action 4!
    98. GoGeometry Action 177!
    99. Shearer Action 5!
    100. V. Pantaloni Chord Problem
  6. Prove Similarity Theorems
    1. GoGeometry Action 111!
    2. Bryan Johnson's Triangle Theorem
  7. Parabolas
    1. Special Conic LR Action
    2. Animation 256
    3. Parabola: Geometric Property (I)
    4. Animation 62
    5. Parabola: Geometric Property (II)
    6. Animation 55
    7. Another Parabola Theorem!
    8. Animation 126
    9. Quarter Circle Surprise!
    10. GoGeometry Action 141!
  8. Ellipses and hyperbolas
    1. Ellipse: Reflective Property
    2. Ellipse: Reflective Property
    3. Animation 47
    4. Hyperbola: Reflective Property
    5. Animation 21
  9. Other Geometry Resources
    1. Geometry Resources

1.

Lines and Angles

  • 1. Cut-The-Knot-Action (5)!
  • 2. Animation 214
  • 3. Cut-the-Knot-Action (3)!
  • 4. Animation 188
  • 5. GoGeometry Action 40!
  • 6. Animation 259
  • 7. GoGeometry Action 41!
  • 8. Animation 260
  • 9. GoGeometry Action 58!
  • 10. Animation 278
  • 11. GoGeometry Action 63!
  • 12. GoGeometry Action 65!
  • 13. Animation 285
  • 14. Concave Quadrilateral Craziness! (GoGeometry Action 80)
  • 15. Hexagonal Surprise! (GoGeometry Action 85)
  • 16. GoGeometry Action 123!
  • 17. GoGeometry Action 143!
  • 18. SolveMyMaths Action 2!
  • 19. GoGeometry Action 158!
  • 20. GoGeometry Action 159!
  • 21. GoGeometry Action 168!
  • 22. GoGeometry Action 169!
  • 23. SolveMyMaths Action 7!
  • 24. Regular Polygon Phenomena!
  • 25. Regular Nonagon Simplicity (GoGeometry Action 175!)
  • 26. More Nonagon Wonders (GoGeometry Action 176!)

1.1.

Cut-The-Knot-Action (5)!

Creation of this applet was inspired by a [url=https://twitter.com/CutTheKnotMath/status/839945580088602629]tweet[/url] from [url=https://twitter.com/CutTheKnotMath]Alexander Bogomolny[/url]. [br][br]In the applet below, a [b][color=#6aa84f]regular pentagon[/color][/b] and a [color=#bf9000][b]regular decagon[/b][/color] share a common side. [br][br][color=#ff00ff][b]What is the measure of the pink angle? [/b][/color] [br][br][color=#0000ff][b]How can you formally prove what this applet informally illustrates? [/b][/color]
Quick (Silent) Demo
– Tim Brzezinski

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2.

Triangles

  • 1. Medians & Equal Areas!
  • 2. Animation 74
  • 3. Animation 90
  • 4. Euler Line (Informal Investigation)
  • 5. Euler Line Generator
  • 6. Morley Action!
  • 7. Animation 113
  • 8. Napoleon Motion!
  • 9. Animation 14
  • 10. Hexagonal Napoleon Theorem?
  • 11. Animation 172
  • 12. Viviani Action! (REVAMPED)
  • 13. Equilateral Action (1)!
  • 14. Animation 91
  • 15. Equilateral Action (2)!
  • 16. Animation 94
  • 17. Equilateral Action (3)!
  • 18. Animation 107
  • 19. Equilateral Action (4)!
  • 20. Animation 108
  • 21. Equilateral Triangle and Rectangle (Area Problem)
  • 22. Animation 154
  • 23. Equal Triangular Areas in a Regular Pentagon
  • 24. Animation 194
  • 25. NCTM Calendar Problem (11-2-2016)
  • 26. Animation 139
  • 27. 9 Point Circle Action
  • 28. Animation 34
  • 29. 9-Point Circle Action (Part 2)
  • 30. Animation 106
  • 31. 9-Point Circle Action (Part 3A)
  • 32. 9-Point Circle (Informal Investigation)
  • 33. Animation 71
  • 34. 9-Point Circle Action (Part 3B)
  • 35. 9-Point Circle Action (Part 4)
  • 36. Animation 53
  • 37. 9-Point Circle Action (Part 5)
  • 38. Animation 115
  • 39. Isosceles Triangle: Surprising Theorem!
  • 40. Animation 121
  • 41. Cut-The-Knot Action !!!
  • 42. More Triangle Action!!!
  • 43. Animation 163
  • 44. Triangle Median & Circles Action!!!
  • 45. Animation 164
  • 46. Desargues Action!!!
  • 47. Animation 209
  • 48. GoGeometry Action 10!
  • 49. Animation 219
  • 50. GoGeometry Action 14!
  • 51. Animation 223
  • 52. GoGeometry Action 17!
  • 53. Animation 228
  • 54. GoGeometry Action 18!
  • 55. Animation 229
  • 56. Cut-The-Knot-Action 8!
  • 57. Animation 230
  • 58. GoGeometry Action 20!
  • 59. Animation 233
  • 60. Cut-The-Knot Action 9!
  • 61. Animation 234
  • 62. Carnot Action! = GoGeometry Action 27!
  • 63. Animation 241
  • 64. GoGeometry Action 29!
  • 65. Animation 244
  • 66. GoGeometry Action 30!
  • 67. Animation 245
  • 68. Incircle + Excircle = GoGeometry Action 31!
  • 69. Animation 246
  • 70. GoGeometry Action 32!
  • 71. Animation 247
  • 72. Cut-The-Knot Action 10!
  • 73. Animation 248
  • 74. GoGeometry Action 34!
  • 75. Animation 250
  • 76. Cut-The-Knot-Action 12!
  • 77. Animation 251
  • 78. GoGeometry Action 35!
  • 79. Animation 252
  • 80. GoGeometry Action 37!
  • 81. Animation 254
  • 82. GoGeometry Action 38!
  • 83. Animation 255
  • 84. GoGeometry Action 39!
  • 85. Animation 258
  • 86. GoGeometry Action 43!
  • 87. Animation 263
  • 88. GoGeometry Action 44!
  • 89. Animation 264
  • 90. GoGeometry Action 51!
  • 91. Animation 270
  • 92. GoGeometry Action 53!
  • 93. Animation 272
  • 94. Euler Line Surprise! = GoGeometry Action 54
  • 95. Animation 273!
  • 96. GoGeometry Action 60!
  • 97. Animation 280
  • 98. Animation 283
  • 99. GoGeometry Action 64!
  • 100. Animation 284
  • 101. GoGeometry Action 66!
  • 102. Animation 286
  • 103. GoGeometry Action 68!
  • 104. Animation 288
  • 105. GoGeometry Action 74!
  • 106. GoGeometry Action 75!
  • 107. GoGeometry Action 76!
  • 108. GoGeometry Action 78!
  • 109. GoGeometry Action 82!
  • 110. GoGeometry Action 83!
  • 111. Cut-the-Knot Action 15!
  • 112. Equilateral Triangle + Semicircle = ? (GoGeometry Action 88)
  • 113. GoGeometry Action 89!
  • 114. GoGeometry Action 90!
  • 115. GoGeometry Action 91!
  • 116. GoGeometry Action 93!
  • 117. GoGeometry Action 97!
  • 118. GoGeometry Action 101 !
  • 119. GoGeometry Action 103!
  • 120. GoGeometry Action 104!
  • 121. GoGeometry Action 105!
  • 122. GoGeometry Action 106!
  • 123. GoGeometry Action 107!
  • 124. GoGeometry Action 109!
  • 125. SolveMyMaths Action 1!
  • 126. GoGeometry Action 110!
  • 127. GoGeometry Action 112!
  • 128. GoGeometry Action 113!
  • 129. GoGeometry Action 115!
  • 130. GoGeometry Action 117!
  • 131. Blanchet Action = GoGeometry Action 118!
  • 132. GoGeometry Action 120!
  • 133. GoGeometry Action 121!
  • 134. GoGeometry Action 124!
  • 135. GoGeometry Action 125!
  • 136. Cut-The-Knot Action 18!
  • 137. Cut-The-Knot Action 19!
  • 138. GoGeometry Action 127!
  • 139. Incircle Extension = GoGeometry Action 132!
  • 140. Incircle Extension 2 = GoGeometry Action 133!
  • 141. GoGeometry Action 134!
  • 142. GoGeometry Action 135!
  • 143. GoGeometry Action 138!
  • 144. GoGeometry Action 147!
  • 145. GoGeometry Action 152!
  • 146. GoGeometry Action 156!
  • 147. GoGeometry Action 160!
  • 148. Regular Decagon Magic!
  • 149. GoGeometry Action 161!
  • 150. GoGeometry Action 162!
  • 151. GoGeometry Action 163!
  • 152. Equilateral Triangles Problem
  • 153. GoGeometry Action 166!
  • 154. Feuerbach Action! (GoGeometry Action 167)
  • 155. GoGeometry Action 170!
  • 156. GoGeometry Action 171!
  • 157. GoGeometry Action 174!
  • 158. GoGeometry Action 182!
  • 159. GoGeometry Action 183!
  • 160. GoGeometry Action 184!
  • 161. GoGeometry Action 185!
  • 162. GoGeometry Action 188!
  • 163. GoGeometry Action 189!
  • 164. Regular Octagon Phenomenon (GoGeometry Action 190)

2.1.

Medians & Equal Areas!

In the app below, the LARGE VERTICES are moveable. The smaller points shown on the triangle's sides are midpoints. Interact with the app below and observe what happens. [br][br]Be sure to move the triangle's large vertices around as you explore!
How would you describe the first three segments drawn? What are they? How do you know this?
What does this animation suggest about all six non-overlapping triangles? Explain how this animation helps suggest your assertion.
– Tim Brzezinski

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2.153.

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2.154.

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2.155.

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2.156.

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2.157.

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2.158.

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2.159.

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2.160.

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2.161.

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2.162.

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2.163.

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2.164.

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3.

Parallelograms

  • 1. Finsler-Hadwiger Action!!!
  • 2. Animation 110
  • 3. Thebault Action!!!
  • 4. Animation 83
  • 5. Van Aubel Action!!!
  • 6. Animation 109
  • 7. Square Problem
  • 8. 3 Squares Problem
  • 9. Animation 120
  • 10. Harmonic Mean In an Isosceles Trapezoid
  • 11. Square Action! (Proof Problem 1)
  • 12. Cut-The-Knot Action (4)!
  • 13. Animation 213
  • 14. Rhombus Action + Sequel = GoGeometry Action 7!
  • 15. Animation 215
  • 16. GoGeometry Action 11!
  • 17. Animation 220
  • 18. GoGeometry Action 12!
  • 19. Animation 221
  • 20. GoGeometry Action 18!
  • 21. Animation 231
  • 22. GoGeometry Action 21!
  • 23. Animation 235
  • 24. GoGeometry Action 22!
  • 25. Animation 236
  • 26. GoGeometry Action 23!
  • 27. Animation 237
  • 28. GoGeometry Action 24!
  • 29. Animation 238
  • 30. GoGeometry Action 36!
  • 31. Animation 253
  • 32. Cut-the-Knot-Action 13!
  • 33. Animation 257
  • 34. GoGeometry Action 42!
  • 35. Animation 261
  • 36. Cut-The-Knot-Action 14!
  • 37. Animation 262
  • 38. GoGeometry Action 45!
  • 39. Animation 265
  • 40. GoGeometry Action 46!
  • 41. Animation 266
  • 42. GoGeometry Action 47!
  • 43. Animation 267
  • 44. Rhombus Surprise! (GoGeometry Action 57)
  • 45. Animation 277
  • 46. GoGeometry Action 67!
  • 47. Animation 287
  • 48. GoGeometry Action 69!
  • 49. Square + Rhombus = Constant Surprise! (GoGeometry Action 71)
  • 50. GoGeometry Action 73!
  • 51. Parallelogram? Why So? (GoGeometry Action 77)
  • 52. Square and Quarter Circle Action = GoGeometry Action 81!
  • 53. 3 Squares Problem (II)
  • 54. GoGeometry Action 94!
  • 55. GoGeometry Action 95!
  • 56. GoGeometry Action 96!
  • 57. GoGeometry Action 98!
  • 58. GoGeometry Action 99!
  • 59. GoGeometry Action 102 !
  • 60. Square + Regular Hexagon Action!
  • 61. GoGeometry Action 108!
  • 62. GoGeometry Action 114!
  • 63. GoGeometry Action 119!
  • 64. Cut-The-Knot Action 17!
  • 65. GoGeometry Action 126!
  • 66. GoGeometry Action 130!
  • 67. GoGeometry Action 136!
  • 68. GoGeometry Action 139!
  • 69. GoGeometry Action 140!
  • 70. Parallelogram + 2 Equilateral Triangles: How to Prove?
  • 71. GoGeometry Action 153!
  • 72. GoGeometry Action 165!
  • 73. Shearer Action 1! (Regular Octagon Problem)
  • 74. Shearer Action 6!
  • 75. GoGeometry Action 181!
  • 76. GoGeometry Action 186!
  • 77. GoGeometry Action 187!

3.1.

Finsler-Hadwiger Action!!!

[color=#000000]In the applet below, simply slide the slider very slowly and enjoy the phenomena you witness. [br][br]After doing so, feel free to adjust the [b][color=#38761D]green [/color][/b]and [b][color=#BF9000]yellow[/color][/b] sliders to change the sizes of the [b][color=#38761D]green [/color][/b]and [b][color=#BF9000]yellow[/color][/b] squares, respectively. You can also change the locations of any of the white points. [br][br]Interact with this applet for a few minutes. Then answer the questions that follow. [/color][br][br]
Write the phenomena you've witnessed several times as a conditional ("if-then") statement.
Can you use coordinate geometry to formally prove what this applet informally illustrates? [br](For starters, why not let the common vertex be (0,0) and go from there?)
– Tim Brzezinski

3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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3.7.

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3.8.

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3.9.

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3.10.

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3.11.

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3.12.

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3.13.

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3.14.

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3.15.

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3.16.

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3.17.

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3.18.

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3.19.

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3.20.

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3.21.

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3.22.

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3.23.

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3.24.

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3.25.

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3.26.

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3.27.

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3.28.

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3.29.

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3.30.

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3.31.

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3.32.

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3.33.

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3.34.

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3.35.

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3.36.

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3.37.

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3.38.

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3.39.

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3.40.

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3.41.

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3.42.

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3.43.

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3.44.

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3.45.

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3.46.

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3.47.

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3.48.

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3.49.

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3.50.

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3.51.

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3.52.

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3.53.

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3.54.

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3.55.

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3.56.

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3.57.

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3.58.

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3.59.

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3.60.

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3.61.

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3.62.

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3.63.

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3.64.

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3.65.

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3.66.

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3.67.

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3.68.

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3.69.

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3.70.

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3.71.

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3.72.

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3.73.

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3.74.

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3.75.

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3.76.

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3.77.

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4.

Geometric Constructions

  • 1. Bizzare Trisection?
  • 2. Animation 198
  • 3. Bizarre Trisection (V2) = GoGeometry Action 145!
  • 4. Angle Trisection by Paper Folding!
  • 5. Bizarre Trisection (V3) = GoGeometry Action 146!

4.1.

Bizzare Trisection?

[color=#000000]The following applet illustrates an alternate means to [b][color=#0000ff]trisect a segment [/color][/b]without using parallel lines. [br][br][b][color=#0000ff]KEY QUESTION(S): [/color][/b][br]Why does this method work? [br]What previously learned theorem(s) justify your conclusions?[br]Explain. [/color]
Quick (Silent) Demo
– Tim Brzezinski

4.2.

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4.3.

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4.4.

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4.5.

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5.

Circle Theorems

  • 1. Butterfly Theorem Action!
  • 2. Animation 132
  • 3. Not Your Everyday Chord & Tangent Theorem
  • 4. Animation 133
  • 5. Cut-The-Knot Action (2) !!!
  • 6. Animation 152
  • 7. Int+Ext Tangent Action!
  • 8. Animation 155
  • 9. Square, Tangent, and Two Congruent & Tangent Circles
  • 10. Animation 153
  • 11. GoGeometry Action 1!
  • 12. Animation 205
  • 13. GoGeometry Action 2!
  • 14. Animation 206
  • 15. GoGeometry Action 3!
  • 16. Animation 207
  • 17. GoGeometry Action 4!
  • 18. Animation 208
  • 19. GoGeometry Action 5!
  • 20. Animation 210
  • 21. GoGeometry Action 6!
  • 22. Animation 212
  • 23. GoGeometry Action 8!
  • 24. Animation 216
  • 25. Cut-The-Knot Action (6)!
  • 26. Animation 217
  • 27. GoGeometry Action 9!
  • 28. Animation 218
  • 29. GoGeometry Action 13!
  • 30. Animation 222
  • 31. GoGeometry Action 15!
  • 32. Animation 224
  • 33. GoGeometry Action 16!
  • 34. Animation 226
  • 35. Cut-The-Knot Action 7!
  • 36. Animation 227
  • 37. GoGeometry Action 19!
  • 38. Animation 232
  • 39. Thales Action + Sequel = GoGeometry Action 25!
  • 40. Animation 239
  • 41. GoGeometry Action 26!
  • 42. Animation 240
  • 43. Carnot Action! = GoGeometry Action 27!
  • 44. Animation 241
  • 45. Incircle + Excircle = GoGeometry Action 31!
  • 46. Animation 246
  • 47. Cut-The-Knot Action 10!
  • 48. Animation 248
  • 49. Cut-The-Knot-Action 11!
  • 50. Animation 249
  • 51. GoGeometry Action 48!
  • 52. Animation 268
  • 53. GoGeometry Action 49!
  • 54. GoGeometry Action 50!
  • 55. Animation 269
  • 56. GoGeometry Action 52!
  • 57. Animation 271
  • 58. 60-Degree Sector Surprise! (GoGeometry Action 55)
  • 59. Animation 274
  • 60. Neusis Logic Action = GoGeometry Action 56!
  • 61. Animation 275
  • 62. Animation 276
  • 63. GoGeometry Action 59!
  • 64. Animation 279
  • 65. GoGeometry Action 61!
  • 66. Animation 281
  • 67. GoGeometry Action 62!
  • 68. Animation 282
  • 69. GoGeometry Action 70!
  • 70. GoGeometry Action 72!
  • 71. GoGeometry Action 79!
  • 72. Geometric Mean Madness! (GoGeometry Action 84)
  • 73. Mickey Mouse Theorem! (Cut-the-Knot Action 16)
  • 74. More Geometric Mean Action! (GoGeometry Action 86)
  • 75. GoGeometry Action 87!
  • 76. GoGeometry Action 92!
  • 77. A Stronger Conclusion!
  • 78. GoGeometry Action 100!
  • 79. GoGeometry Action 116!
  • 80. GoGeometry Action 122!
  • 81. GoGeometry Action 128!
  • 82. GoGeometry Action 129!
  • 83. GoGeometry Action 131!
  • 84. GoGeometry Action 137!
  • 85. GoGeometry Action 142!
  • 86. Circle Optimization Challenge!
  • 87. SolveMyMaths Action 5!
  • 88. GoGeometry Action 148!
  • 89. Cyclic Quadrilateral: Quick Warm Up (GoGeometry Action 149!)
  • 90. Archimedes Insight = GoGeometry Action 150!
  • 91. More Archimedes Insight = GoGeometry Action 151!
  • 92. GoGeometry Action 154!
  • 93. GoGeometry Action 155!
  • 94. Archimedes Insight 3: GoGeometry Action 157!
  • 95. GoGeometry Action 164!
  • 96. GoGeometry Action 172!
  • 97. Shearer Action 4!
  • 98. GoGeometry Action 177!
  • 99. Shearer Action 5!
  • 100. V. Pantaloni Chord Problem

5.1.

Butterfly Theorem Action!

[color=#cc0000][b]Note: [/b][/color][br]Creation of this applet was inspired by a [url=https://twitter.com/CutTheKnotMath/status/791398769459924998]tweet [/url]from [url=https://twitter.com/CutTheKnotMath]Alexander Bogomolny[/url] (at [url=http://www.cut-the-knot.org/]Cut-the-Knot[/url].) [br][br]The applet below dynamically illustrates a theorem known as the [b]Butterfly Theorem[/b]. [br][br]Interact with this applet for a few minutes. As you do, feel free to change the location(s) of any 1 (or more) of the [color=#674ea7][b]BIG POINT(S) that are already there[/b][/color][color=#274e13][b] (or will soon appear)[/b][/color]! [br][br]How can you formally prove what this applet informally illustrates?
– Tim Brzezinski

5.2.

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5.3.

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5.4.

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5.5.

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5.6.

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5.7.

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5.8.

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5.9.

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5.10.

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5.11.

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5.12.

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5.13.

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5.14.

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5.15.

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5.16.

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5.17.

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5.18.

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5.19.

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5.20.

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5.21.

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5.22.

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5.23.

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5.24.

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5.25.

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5.26.

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5.27.

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5.28.

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5.29.

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5.30.

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5.31.

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5.32.

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5.33.

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5.34.

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5.35.

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5.36.

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5.37.

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5.38.

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5.39.

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5.40.

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5.41.

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5.42.

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5.43.

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5.44.

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5.45.

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5.46.

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5.47.

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5.48.

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5.49.

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5.50.

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5.51.

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5.52.

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5.53.

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5.54.

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5.55.

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5.56.

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5.57.

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5.58.

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5.59.

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5.60.

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5.61.

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5.62.

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5.63.

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5.64.

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5.65.

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5.66.

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5.67.

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5.68.

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5.69.

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5.70.

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5.71.

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5.72.

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5.73.

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5.74.

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5.75.

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5.76.

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5.77.

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5.78.

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5.79.

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5.80.

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5.81.

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5.82.

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5.83.

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5.84.

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5.85.

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5.86.

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5.87.

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5.88.

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5.89.

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5.90.

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5.91.

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5.92.

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5.93.

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5.94.

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5.95.

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5.96.

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5.97.

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5.98.

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5.99.

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5.100.

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6.

Prove Similarity Theorems

  • 1. GoGeometry Action 111!
  • 2. Bryan Johnson's Triangle Theorem

6.1.

GoGeometry Action 111!

Creation of this applet was inspired by a [url=https://twitter.com/gogeometry/status/942037810378366977]tweet[/url] from [url=https://twitter.com/gogeometry]Antonio Gutierrez[/url] (GoGeometry). [br][br]Note: This applet works best if the quadrilateral is kept convex. [br][br][b][color=#ff00ff]How can we formally prove the phenomena dynamically illustrated here? [/color][/b]
Quick (Silent) Demo
– Tim Brzezinski

6.2.

–

7.

Parabolas

  • 1. Special Conic LR Action
  • 2. Animation 256
  • 3. Parabola: Geometric Property (I)
  • 4. Animation 62
  • 5. Parabola: Geometric Property (II)
  • 6. Animation 55
  • 7. Another Parabola Theorem!
  • 8. Animation 126
  • 9. Quarter Circle Surprise!
  • 10. GoGeometry Action 141!

7.1.

Special Conic LR Action

A conic section is shown below. [br][br][b]This conic's vertex is black.[/b][br][b][color=#999999]Its directrix is gray.[/color][/b][br][b][color=#1e84cc]Its focus is blue. [br][/color][color=#ff7700]The orange point is a point that lies on this conic section itself. [br][/color][/b][br]Interact with this applet for a couple of minutes.[br]Then answer the questions that follow. [br][br]
1.
What type of conic is illustrated above? Explain how/why you know this to be true.
2.
A focal chord of this particular conic is defined as a segment that passes through this conic's [b][color=#1e84cc]focus [/color][/b]AND whose endpoints lie on the conic itself. How would you describe the focal chord seen with respect to this conic's axis of symmetry? (Hint: How would you describe the intersection?)
3.
A focal chord of this conic with the type of intersection described in (2) above is said to be a LATUS RECTUM of this conic. How does the length of the latus rectum compare with the [b]distance from this conic's vertex to [color=#1e84cc]focus?[/color][/b]
4.
Formally prove your response to (3) true. [br]For simplicity, feel free to place the [b]vertex at (0, 0)[/b] and the [b][color=#1e84cc]focus at (0, p). [/color][/b]
Quick (Silent) Demo
– Tim Brzezinski

7.2.

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7.3.

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7.4.

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7.5.

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7.6.

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7.7.

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7.8.

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7.9.

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7.10.

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8.

Ellipses and hyperbolas

  • 1. Ellipse: Reflective Property
  • 2. Ellipse: Reflective Property
  • 3. Animation 47
  • 4. Hyperbola: Reflective Property
  • 5. Animation 21

8.1.

Ellipse: Reflective Property

The applet below contains an ellipse with both its foci shown. [b]Feel free to place point [i]P, [/i][color=#ff7700]the orange point[/color], and both foci of the ellipse anywhere you'd like at any time![/b][br][br]How would you describe the phenomena you see?
[i]Do a little exploratory research on the following topics:[br]Specifically, how are ellipses used in each application? [/i][br][br][b]1) Whispering Galleries[br]2) Elliptic Lithotripsy (Destroying Kidney Stones) [br][/b][br][color=#0000ff][b]How is the principle illustrated in the applet above "reflected" (no pun intended) in each of these applications? [/b][/color]
– Tim Brzezinski

8.2.

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8.3.

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8.4.

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8.5.

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9.

Other Geometry Resources

  • 1. Geometry Resources

9.1.

Geometry Resources

[list][*][b][url=https://www.geogebra.org/m/z8nvD94T]Congruence (Volume 1)[/url][/b][/*][*][b][url=https://www.geogebra.org/m/munhXmzx]Congruence (Volume 2)[/url][/b][/*][*][b][url=https://www.geogebra.org/m/dPqv8ACE]Similarity, Right Triangles, Trigonometry[/url][/b][/*][*][b][url=https://www.geogebra.org/m/C7dutQHh]Circles[/url][/b][/*][*][b][url=https://www.geogebra.org/m/K2YbdFk8]Coordinate and Analytic Geometry[/url][/b][/*][*][b][url=https://www.geogebra.org/m/xDNjSjEK]Area, Surface Area, Volume, 3D, Cross Section[/url] [/b][/*][*][b][url=https://www.geogebra.org/m/NjmEPs3t]Proof Challenges[/url]  [/b][/*][/list]
What phenomenon is dynamically being illustrated here? (Vertices are moveable.)
What phenomenon is dynamically being illustrated here? (Vertices are moveable.)
– Tim Brzezinski

Information