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拖动点与滑块探索几何定理

拖动点与滑块探索几何定理
几何画板及绘图内容加载中，请稍等……
2015-02-22 02:15
测试改名0圆三角函数几何作图
所属专题 Proofs Without Words Proofs Without Words
①本范例通过拖动点与滑块，演示三角形内切圆与相关几何性质；②涉及核心数学概念包括三角形、内切圆、几何作图、三角函数等；③构造中运用了Point, CircleArc, Incircle, Midpoint, Segment, OrthogonalLine, Angle, Rotate等GeoGebra命令与对象；④适合高中及以上学生用于探索几何定理与深化对几何作图的理解。

GeoGebra 命令及参数

  1. Point(xAxis) → A
  2. Intersect(xAxis, yAxis) → B
  3. CircleArc(B, D, A) → c
  4. Point(c) → E
  5. Polygon(A, E, 4) → poly1, a, b, d, e, F, G
  6. Polygon(E, C, 4) → poly2, f, g, h, i, H, I
  7. Polygon(C, A, 4) → poly3, j, k, l, m, J, K
  8. Incircle(A, E, C) → p
  9. Intersect(p, f) → L, P
  10. Intersect(p, a) → M,
  11. Intersect(p, j) → N, O
  12. Midpoint(F, A) → Q
  13. Midpoint(E, H) → R
  14. Mirror(M, Q) → M'
  15. Mirror(L, R) → L'
  16. Segment(L, L') → n
  17. OrthogonalLine(R, n) → q
  18. Segment(M', M) → r
  19. OrthogonalLine(Q, r) → s
  20. Intersect(s, e) → S
  21. Intersect(s, b) → T
  22. Intersect(q, i) → U
  23. Intersect(q, g) → V
  24. Polygon(P, V, R) → poly4, r_1, p_1, v
  25. Polygon(P, U, R) → poly5, r_2, p_2, u
  26. Polygon(U, R, L') → poly6, l', u_1, r_3
  27. Polygon(R, V, L') → poly7, l'_1, r_4, v_1
  28. Polygon(U, I, L') → poly8, l'_2, u_2, i_1
  29. Polygon(U, E, P) → poly9, p_3, u_3, e_1
  30. Polygon(L', H, V) → poly10, v_2, l'_3, h_1
  31. Polygon(C, V, P) → poly11, p_4, c_1, v_3
  32. Polygon(S, M, T, M') → poly12, s_1, m_1, t, m'
  33. Polygon(S, A, M) → poly13, m_2, s_2, a_1
  34. Polygon(T, E, M) → poly14, m_3, t_1, e_2
  35. Polygon(T, M', F) → poly15, f_1, t_2, m'_1
  36. Polygon(G, M', S) → poly16, s_3, g_1, m'_2
  37. Angle(N, A, M) → α
  38. Angle(P, C, N) → β
  39. If(0 ≤ o ≤ 1, 1 - o, If[o > 1, 0, 1]) → d1
  40. Dilate(M, d1, K) → M'_1
  41. Dilate(S, d1, J) → S'
  42. Dilate(M', d1, A) → M''
  43. Dilate(T, d1, C) → T'
  44. Vector(M, M'_1) → w
  45. Vector(S, S') → w_1
  46. Vector(M', M'') → b_1
  47. Vector(T, T') → d_1
  48. Translate(S, w) → S'_1
  49. Translate(A, w) → A'
  50. Translate(M, w) → M'_2
  51. Polygon(S'_1, A', M'_2) → poly13', m'_3, s'_1, a'
  52. If(0 ≤ o ≤ 1, o - 0, If[o > 1, 1, 0]) → r1
  53. Rotate(S'_1, (-((α * r1))), M'_1) → W
  54. Rotate(A', (-((α * r1))), M'_1) → A''
  55. Rotate(M'_2, (-((α * r1))), M'_1) → Z
  56. Polygon(W, A'', Z) → poly13'', z_1, w_2, a''
  57. Translate(G, w_1) → G'
  58. Translate(M', w_1) → M''_1
  59. Translate(S, w_1) → S'_2
  60. Polygon(G', M''_1, S'_2) → poly16', s'_2, g', m''_1
  61. Translate(T, b_1) → T'_1
  62. Translate(M', b_1) → M''_2
  63. Translate(F, b_1) → F'
  64. Polygon(T'_1, M''_2, F') → poly15', f', t'_1, m''_2
  65. Translate(T, d_1) → T'_2
  66. Translate(E, d_1) → E'
  67. Translate(M, d_1) → M'_3
  68. Polygon(T'_2, E', M'_3) → poly14', m'_4, t'_2, e'
  69. Rotate(G', (-((r1 * α))), S') → G''
  70. Rotate(M''_1, (-((r1 * α))), S') → A_1
  71. Rotate(S'_2, (-((r1 * α))), S') → B_1
  72. Polygon(G'', A_1, B_1) → poly16'', b_2, g'', a_2
  73. Rotate(T'_1, (-((r1 * α))), M'') → C_1
  74. Rotate(M''_2, (-((r1 * α))), M'') → D_1
  75. Rotate(F', (-((r1 * α))), M'') → F''
  76. Polygon(C_1, D_1, F'') → poly15'', f'', c_2, d_2
  77. Rotate(T'_2, (-((r1 * α))), T') → E_1
  78. Rotate(E', (-((r1 * α))), T') → E''
  79. Rotate(M'_3, (-((r1 * α))), T') → F_1
  80. Polygon(E_1, E'', F_1) → poly14'', f_2, e_3, e''
  81. If(1 ≤ o ≤ 2, 2 - o, If[o > 2, 0, 1]) → d2
  82. Dilate(E, d2, G'') → E'_1
  83. Dilate(C, d2, A'') → C'
  84. Dilate(H, d2, E'') → H'
  85. Dilate(I, d2, N) → I'
  86. Vector(E, E'_1) → j_1
  87. Vector(I, I') → k_1
  88. Vector(H, H') → l_1
  89. Vector(C, C') → n_1
  90. Translate(U, j_1) → U'
  91. Translate(E, j_1) → E'_2
  92. Translate(P, j_1) → P'
  93. Polygon(U', E'_2, P') → poly9', p', u', e'_2
  94. If(1 ≤ o ≤ 2, o - 1, If[o > 2, 1, 0]) → r2
  95. Rotate(U', (r2 * β), E'_1) → U''
  96. Rotate(E'_2, (r2 * β), E'_1) → G_1
  97. Rotate(P', (r2 * β), E'_1) → P''
  98. Polygon(U'', G_1, P'') → poly9'', p'', u'', g_2
  99. Translate(U, k_1) → U'_1
  100. Translate(I, k_1) → I'_1
  101. Translate(L', k_1) → L''
  102. Polygon(U'_1, I'_1, L'') → poly8', l'', u'_1, i'_1
  103. Rotate(U'_1, (r2 * β), I') → H_1
  104. Rotate(I'_1, (r2 * β), I') → I_1
  105. Rotate(L'', (r2 * β), I') → L'''
  106. Polygon(H_1, I_1, L''') → poly8'', l''', h_2, i_2
  107. Translate(L', l_1) → L''_1
  108. Translate(H, l_1) → H'_1
  109. Translate(V, l_1) → V'
  110. Polygon(L''_1, H'_1, V') → poly10', v', l''_1, h'_1
  111. Rotate(L''_1, (r2 * β), H') → J_1
  112. Rotate(H'_1, (r2 * β), H') → K_1
  113. Rotate(V', (r2 * β), H') → V''
  114. Polygon(J_1, K_1, V'') → poly10'', v'', j_2, k_2
  115. Translate(C, n_1) → C'_1
  116. Translate(V, n_1) → V'_1
  117. Translate(P, n_1) → P'_1
  118. Polygon(C'_1, V'_1, P'_1) → poly11', p'_1, c'_1, v'_1
  119. Rotate(C'_1, (r2 * β), C') → L_1
  120. Rotate(V'_1, (r2 * β), C') → M_1
  121. Rotate(P'_1, (r2 * β), C') → N_1
  122. Polygon(L_1, M_1, N_1) → poly11'', n_2, l_2, m_4
  123. If(2 ≤ o ≤ 3, 3 - o, If[o > 3, 0, 1]) → d3
  124. Dilate(P, d3, A_1) → P'_2
  125. Dilate(V, d3, W) → V'_2
  126. Vector(P, P'_2) → q_1
  127. Translate(P, q_1) → P'_3
  128. Translate(U, q_1) → U'_2
  129. Translate(R, q_1) → R'
  130. Polygon(P'_3, U'_2, R') → poly5', r', p'_3, u'_2
  131. If(2 ≤ o ≤ 3, o - 2, If[o > 3, 1, 0]) → r3
  132. Rotate(P'_3, (r3 * β), P'_2) → O_1
  133. Rotate(U'_2, (r3 * β), P'_2) → P_1
  134. Rotate(R', (r3 * β), P'_2) → R''
  135. Polygon(O_1, P_1, R'') → poly5'', r'', o_1, p_5
  136. Vector(V, V'_2) → q_2
  137. Translate(P, q_2) → P'_4
  138. Translate(V, q_2) → V'_3
  139. Translate(R, q_2) → R'_1
  140. Polygon(P'_4, V'_3, R'_1) → poly4', r'_1, p'_4, v'_3
  141. Rotate(P'_4, (r3 * β), V'_2) → Q_1
  142. Rotate(V'_3, (r3 * β), V'_2) → R_1
  143. Rotate(R'_1, (r3 * β), V'_2) → S_1
  144. Polygon(Q_1, R_1, S_1) → poly4'', s_4, q_3, r_5
  145. Dilate(L', d3, F_1) → L''_2
  146. Vector(L', L''_2) → w_3
  147. Translate(R, w_3) → R'_2
  148. Translate(V, w_3) → V'_4
  149. Translate(L', w_3) → L''_3
  150. Polygon(R'_2, V'_4, L''_3) → poly7', l''_3, r'_2, v'_4
  151. Rotate(R'_2, (r3 * β), L''_2) → T_1
  152. Rotate(V'_4, (r3 * β), L''_2) → U_1
  153. Rotate(L''_3, (r3 * β), L''_2) → V_1
  154. Polygon(T_1, U_1, V_1) → poly7'', v_4, t_3, u_4
  155. Dilate(U, d3, C_1) → U'_3
  156. Vector(U, U'_3) → a_3
  157. Translate(U, a_3) → U'_4
  158. Translate(R, a_3) → R'_3
  159. Translate(L', a_3) → L''_4
  160. Polygon(U'_4, R'_3, L''_4) → poly6', l''_4, u'_4, r'_3
  161. Rotate(U'_4, (r3 * β), U'_3) → W_1
  162. Rotate(R'_3, (r3 * β), U'_3) → Z_1
  163. Rotate(L''_4, (r3 * β), U'_3) → A_2
  164. Polygon(W_1, Z_1, A_2) → poly6'', a_4, w_4, z_2
  165. If(3 ≤ o ≤ 4, 4 - o, If[o > 4, 0, 1]) → d4
  166. If(3 ≤ o ≤ 4, o - 3, If[o > 4, 1, 0]) → r4
  167. Dilate(S, d4, T_1) → S'_3
  168. Vector(S, S'_3) → b_3
  169. Translate(S, b_3) → S'_4
  170. Translate(M, b_3) → M'_4
  171. Translate(T, b_3) → T'_3
  172. Translate(M', b_3) → M''_3
  173. Polygon(S'_4, M'_4, T'_3, M''_3) → poly12', s'_4, m'_5, t'_3, m''_3
  174. Rotate(S'_4, (r4 * β), S'_3) → B_2
  175. Rotate(M'_4, (r4 * β), S'_3) → C_2
  176. Rotate(T'_3, (r4 * β), S'_3) → D_2
  177. Rotate(M''_3, (r4 * β), S'_3) → E_2
  178. Polygon(B_2, C_2, D_2, E_2) → poly12'', b_4, c_3, d_3, e_4

AI 绘图提示语参考

绘制一个三角形ABC，通过拖动点和滑块探索其内切圆的性质。首先在x轴上创建点A，原点B，然后以B为中心画弧交x轴于D，再创建点E，使用E、C、A三点构造多边形。接着构造三角形ABC的内切圆p，并找到其与多边形边的交点L、M、N、O。通过中点和镜像操作，构造一系列多边形和线段，最终形成复杂的几何图形，探索三角形和内切圆的关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。