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凯西定理演示

凯西定理演示
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2019-12-08 20:22
圆角三角函数几何作图
①本范例演示了凯西定理在几何中的构造与应用，该定理涉及圆与三角形的特殊关系；②核心数学概念包括圆、相似变换、三角形相似、切线、单位圆等；③作图要点运用了Intersect（交点）、Circle（圆）、Point（点）、Segment（线段）、Angle（角度）、Min（最小值）、Semicircle（半圆）、Line（直线）、OrthogonalLine（垂线）、Element（元素）等GeoGebra命令；④适合高中及以上数学学习者，用于理解几何构造与定理证明。

GeoGebra 命令及参数

  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