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Casey's Theorem Demonstration

Casey's Theorem Demonstration
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2019-12-08 20:22
CircleAngleTrigonometryGeometric Construction
This example demonstrates the geometric construction and application of Casey's theorem, which involves special relationships between circles and triangles. Key math concepts include circles, similarity transformations, triangle similarity, tangent lines, unit circles, etc. GeoGebra commands used in construction are Intersect (intersection), Circle, Point, Segment, Angle, Min, Semicircle, Line, OrthogonalLine, Element. Suitable for high school and above learners to understand geometric constructions and theorem proofs.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, 1) → c
  3. Point(c) → B
  4. Segment(A, B) → a
  5. Point(a) → C
  6. Circle(C, B) → d
  7. Point(c) → D
  8. Angle(B, A, D) → α
  9. Segment(C, B) → r_1
  10. Segment(D, A) → b
  11. Circle(D, s) → e
  12. Intersect(e, b, 1) → E
  13. Circle(E, D) → f
  14. Segment(D, E) → g
  15. Point(g) → F
  16. Circle(F, D) → h
  17. Point(c) → G
  18. Angle(B, A, G) → β_1
  19. Angle(D, A, G) → β_2
  20. Segment(D, F) → r_2
  21. Segment(G, A) → i
  22. Min(t_1, t_2) → t
  23. Circle(G, t) → p
  24. Intersect(p, i, 1) → H
  25. Segment(G, H) → j
  26. Point(j) → I
  27. Circle(I, G) → q
  28. Point(c) → J
  29. Angle(B, A, J) → γ_1
  30. Angle(D, A, J) → γ_2
  31. Angle(G, A, J) → γ_3
  32. Segment(G, I) → r_3
  33. Min(u_1, Min[u_2, u_3]) → u
  34. Segment(A, J) → l
  35. Circle(J, u) → r
  36. Intersect(r, l) → K,
  37. Circle(K, J) → c_1
  38. Segment(J, K) → m
  39. Point(m) → L
  40. Circle(L, J) → d_1
  41. Segment(J, L) → r_4
  42. Semicircle(F, C) → e_1
  43. Semicircle(I, F) → f_1
  44. Semicircle(L, I) → g_1
  45. Semicircle(C, L) → h_1
  46. Circle(C, r_5) → k_1
  47. Circle(F, r_6) → p_1
  48. Circle(I, r_7) → q_1
  49. Intersect(e_1, k_1, 1) → M
  50. Intersect(f_1, p_1, 1) → N
  51. Intersect(g_1, q_1, 1) → O
  52. Line(F, M) → n
  53. OrthogonalLine(F, n) → a_1
  54. OrthogonalLine(C, n) → b_1
  55. Intersect(h, a_1, 2) → P
  56. Intersect(d, b_1, 2) → Q
  57. Line(I, N) → i_1
  58. OrthogonalLine(I, i_1) → j_1
  59. OrthogonalLine(F, i_1) → n_1
  60. Intersect(q, j_1, 2) → R
  61. Intersect(h, n_1, 2) → S
  62. Line(O, L) → s_1
  63. OrthogonalLine(L, s_1) → a_6
  64. OrthogonalLine(I, s_1) → b_2
  65. Intersect(q, b_2, 1) → T
  66. Intersect(d_1, a_6, 1) → U
  67. Segment(P, Q) → c_2
  68. Segment(R, S) → d_2
  69. Segment(T, U) → e_2
  70. Circle(C, r_8) → f_2
  71. Intersect(h_1, f_2, 1) → V
  72. Line(V, L) → g_5
  73. OrthogonalLine(C, g_5) → h_2
  74. OrthogonalLine(L, g_5) → i_2
  75. Intersect(d, h_2, 2) → W
  76. Intersect(d_1, i_2, 2) → Z
  77. Segment(W, Z) → j_2
  78. Circle(C, r_9) → k_2
  79. Circle(F, r_{10}) → p_2
  80. Semicircle(C, I) → q_2
  81. Semicircle(L, F) → s_2
  82. Intersect(k_2, q_2, 1) → A_1
  83. Intersect(p_2, s_2, 1) → B_1
  84. Line(I, A_1) → n_2
  85. OrthogonalLine(I, n_2) → a_3
  86. OrthogonalLine(C, n_2) → b_3
  87. Intersect(q, a_3, 1) → C_1
  88. Intersect(d, b_3, 1) → D_1
  89. Segment(C_1, D_1) → a_2
  90. Line(B_1, L) → d_3
  91. OrthogonalLine(F, d_3) → e_3
  92. OrthogonalLine(L, d_3) → f_3
  93. Intersect(h, e_3, 1) → E_1
  94. Intersect(d_1, f_3, 1) → F_1
  95. Segment(E_1, F_1) → g_3
  96. Circle(F, (-r_5)) → h_3
  97. Circle(I, (-r_6)) → k_3
  98. Circle(L, (-r_7)) → p_3
  99. Intersect(e_1, h_3, 1) → G_1
  100. Line(G_1, C) → i_3
  101. OrthogonalLine(F, i_3) → j_3
  102. OrthogonalLine(C, i_3) → l_3
  103. Intersect(h, j_3, 2) → H_1
  104. Intersect(d, l_3, 2) → I_1
  105. Intersect(f_1, k_3, 1) → J_1
  106. Line(J_1, F) → n_3
  107. OrthogonalLine(F, n_3) → q_3
  108. OrthogonalLine(I, n_3) → s_3
  109. Intersect(q, s_3, 2) → K_1
  110. Intersect(h, q_3, 2) → L_1
  111. Intersect(g_1, p_3, 1) → M_1
  112. Line(I, M_1) → t_3
  113. OrthogonalLine(I, t_3) → a_4
  114. OrthogonalLine(L, t_3) → b_4
  115. Intersect(q, a_4, 1) → N_1
  116. Intersect(d_1, b_4, 1) → O_1
  117. Segment(I_1, H_1) → c_4
  118. Segment(L_1, K_1) → d_4
  119. Segment(N_1, O_1) → e_4
  120. Circle(L, (-r_8)) → f_4
  121. Intersect(h_1, f_4, 1) → P_1
  122. Line(C, P_1) → g_4
  123. OrthogonalLine(C, g_4) → h_4
  124. OrthogonalLine(L, g_4) → i_4
  125. Intersect(d, h_4, 2) → Q_1
  126. Intersect(d_1, i_4, 2) → R_1
  127. Segment(R_1, Q_1) → j_4
  128. Circle(I, (-r_9)) → k_4
  129. Circle(L, (-r_{10})) → p_4
  130. Intersect(q_2, k_4, 1) → S_1
  131. Line(S_1, C) → l_4
  132. OrthogonalLine(I, l_4) → m_4
  133. OrthogonalLine(C, l_4) → n_4
  134. Intersect(q, m_4, 1) → T_1
  135. Intersect(d, n_4, 1) → U_1
  136. Intersect(s_2, p_4, 1) → V_1
  137. Line(F, V_1) → q_4
  138. OrthogonalLine(F, q_4) → s_4
  139. OrthogonalLine(L, q_4) → t_4
  140. Intersect(h, s_4, 1) → W_1
  141. Intersect(d_1, t_4, 1) → Z_1
  142. Segment(T_1, U_1) → a_5
  143. Segment(W_1, Z_1) → g_2
  144. Element(RemoveUndefined[{a_2, a_5}], 1) → a_7
  145. Element(RemoveUndefined[{c_2, c_4}], 1) → c_7
  146. Element(RemoveUndefined[{e_2, e_4}], 1) → e_7
  147. Element(RemoveUndefined[{d_2, d_4}], 1) → d_7
  148. Element(RemoveUndefined[{g_2, g_3}], 1) → g_7
  149. Element(RemoveUndefined[{j_2, j_4}], 1) → j_7
  150. Line(P, Q) → c_3
  151. Line(T, U) → b_5
  152. Line(Z, W) → c_5
  153. Line(D_1, C_1) → d_5
  154. Line(E_1, F_1) → e_5
  155. Line(R, S) → f_5
  156. Line(H_1, I_1) → h_5
  157. Line(K_1, L_1) → i_5
  158. Line(T_1, U_1) → j_5
  159. Line(W_1, Z_1) → k_5
  160. Line(N_1, O_1) → l_5
  161. Line(R_1, Q_1) → m_5