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直角三角形内的三个内切圆半径关系

直角三角形内的三个内切圆半径关系
几何画板及绘图内容加载中，请稍等……
2017-08-15 22:50
测试改名0圆几何作图
该范例展示了直角三角形内三个不同三角形的内切圆半径之间的关系。核心数学概念包括几何均值和圆。通过使用GeoGebra命令如半圆、点、线段、角度、线、正交线、交点、旋转、射线、多边形、放大、最小值、圆弧等，动态地展示了这些关系。适合初中及以上学生学习几何作图和圆的相关性质。

GeoGebra 命令及参数

  1. Semicircle(A, B) → c
  2. Point(c) → C
  3. Segment(A, C) → f
  4. Segment(C, B) → g
  5. Segment(B, A) → h
  6. Angle(A, C, B) → α
  7. Line(C, A) → i
  8. Line(C, B) → j
  9. Line(A, B) → k
  10. OrthogonalLine(C, k) → l
  11. Intersect(k, l) → D
  12. OrthogonalLine(D, j) → m
  13. OrthogonalLine(D, i) → n
  14. Intersect(j, m) → E
  15. Intersect(i, n) → F
  16. Angle(D, E, B) → β
  17. Angle(A, F, D) → γ
  18. Line(F, E) → p
  19. Angle(B, A, C) → δ
  20. Rotate(B, δ / 2, A) → B'
  21. Angle(B, A, B') → ε
  22. Angle(C, B, A) → ζ
  23. Rotate(C, ζ / 2, B) → C'
  24. Angle(C, B, C') → η
  25. Line(D, E) → q
  26. Line(D, F) → r
  27. Ray(B, C') → s
  28. Intersect(q, s) → G
  29. Ray(A, B') → t
  30. Intersect(r, t) → H
  31. Ray(B, G) → a
  32. Line(D, a) → b
  33. Intersect(i, b) → I
  34. Ray(A, H) → d
  35. Intersect(b, d) → J
  36. Line(D, t) → e
  37. Intersect(e, s) → K
  38. Angle(F, E, D) → θ
  39. Angle(D, F, E) → λ
  40. Rotate(E, (-(λ / 2)), F) → E'
  41. Angle(E', F, E) → ω
  42. Ray(F, E') → f_1
  43. Rotate(F, θ / 2, E) → F'
  44. Angle(F, E, F') → κ
  45. Ray(E, F') → g_1
  46. Intersect(f_1, g_1) → L
  47. Angle(C, D, A) → μ
  48. Polygon(A, C, B) → poly1, b_1, a_1, c_1
  49. Dilate(C, If[0 < o ≤ 1, 1 - o, 0], D) → C'_1
  50. Segment(C, C'_1) → h_1
  51. Dilate(D, If[1 < o ≤ 2, 2 - o, 0], F) → D'
  52. Dilate(D, If[1 < o ≤ 2, 2 - o, 0], E) → D'_1
  53. Segment(D, D'_1) → i_1
  54. Segment(D, D') → j_1
  55. Segment(A, J) → k_1
  56. Segment(D, J) → l_1
  57. Segment(D, K) → m_1
  58. Segment(B, K) → n_1
  59. Segment(F, L) → p_1
  60. Segment(E, L) → q_1
  61. Min({k_1, l_1, m_1, n_1, q_1, p_1}) → u
  62. Circle(A, (0.4 * u)) → d_1
  63. Circle(D, (0.4 * u)) → e_1
  64. Circle(B, (0.4 * u)) → r_1
  65. Circle(E, (0.4 * u)) → s_1
  66. Circle(F, (0.4 * u)) → t_1
  67. Intersect(d_1, k) → M, N
  68. Intersect(d_1, k_1) → O,
  69. Intersect(e_1, l_1) → P,
  70. Intersect(e_1, k) → Q, R
  71. Intersect(e_1, m_1) → S,
  72. Intersect(r_1, k) → T, U
  73. Intersect(r_1, n_1) → V,
  74. Intersect(t_1, j_1, 1) → W
  75. Intersect(t_1, p_1, 1) → Z
  76. Intersect(s_1, p, 1) → A_1
  77. Intersect(s_1, q_1, 1) → B_1
  78. CircleSector(A, N, O) → c_2
  79. Intersect(e_1, j_1, 1) → C_1
  80. CircleSector(D, C_1, P) → d_2
  81. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], J) → A'
  82. Dilate(D, If[2 < o ≤ 3, 3 - o, 0], J) → D'_2
  83. Segment(A, A') → f_2
  84. Segment(D, D'_2) → g_2
  85. Dilate(D, If[3 < o ≤ 4, 4 - o, 0], K) → D'_3
  86. Dilate(B, If[3 < o ≤ 4, 4 - o, 0], K) → B'_1
  87. Segment(D, D'_3) → h_2
  88. Segment(B, B'_1) → i_2
  89. Angle(F, D, P) → ϕ
  90. Rotate(d_2, If[4 < o ≤ 5, ((o - 4) * ϕ), ϕ], D) → e_2
  91. Rotate(c_2, If[4 < o ≤ 5, ((o - 4) * ε), ε], A) → k_2
  92. Dilate(D, If[5 < o ≤ 6, 6 - o, 0], B) → D'_4
  93. Vector(D, D'_4) → v
  94. Dilate(A, If[5 < o ≤ 6, 6 - o, 0], D) → A'_1
  95. Vector(A, A'_1) → w
  96. Translate(k_2, w) → p_2
  97. Translate(e_2, v) → q_2
  98. Dilate(F, If[6 < o ≤ 7, 7 - o, 0], E) → F'_1
  99. Segment(F, F'_1) → j_2
  100. CircleSector(F, W, Z) → r_2
  101. CircleSector(E, A_1, B_1) → s_2
  102. Dilate(F, If[7 < o ≤ 8, 8 - o, 0], L) → F'_2
  103. Segment(F, F'_2) → l_2
  104. Dilate(E, If[8 < o ≤ 9, 9 - o, 0], L) → E'_1
  105. Segment(E, E'_1) → m_2
  106. Rotate(r_2, If[9 < o ≤ 10, ((o - 9) * ω), ω], F) → t_2
  107. Rotate(s_2, If[10 < o ≤ 11, ((o - 10) * κ), κ], E) → c_3
  108. OrthogonalLine(K, j) → n_2
  109. OrthogonalLine(J, i) → a_2
  110. OrthogonalLine(L, j_2) → b_2
  111. Intersect(j, n_2) → D_1
  112. Intersect(b_2, j_2) → E_1
  113. Intersect(a_2, i) → F_1
  114. Angle(A, F_1, J) → ι
  115. Angle(F, E_1, L) → ν
  116. Angle(E, D_1, K) → ξ
  117. Dilate(J, If[11 < o ≤ 12, 12 - o, 0], F_1) → J'
  118. Dilate(K, If[11 < o ≤ 12, 12 - o, 0], D_1) → K'
  119. Dilate(L, If[11 < o ≤ 12, 12 - o, 0], E_1) → L'
  120. Segment(J, J') → f_3
  121. Segment(L, L') → g_3
  122. Segment(K, K') → h_3
  123. Circle(L, L') → d_3
  124. Circle(K, K') → e_3
  125. Circle(J, J') → k_3
  126. Dilate(F, If[12 < o ≤ 13, 13 - o, 0], C) → F'_3
  127. Vector(F_1, F'_3) → u_1
  128. Translate(J, u_1) → J'_1
  129. Translate(J', u_1) → J''
  130. Segment(J'_1, J'') → i_3
  131. Dilate(D_1, If[13 < o ≤ 14, 14 - o, 0], J'_1) → G_1
  132. Vector(D_1, G_1) → v_1
  133. Translate(K, v_1) → K'_1
  134. Translate(K', v_1) → K''
  135. Segment(K'_1, K'') → j_3
  136. Rotate(K'_1, If[14 < o ≤ 15, ((o - 14) * π / 2), π / 2], J'_1) → H_1
  137. Rotate(K'', If[14 < o ≤ 15, ((o - 14) * π / 2), π / 2], J'_1) → K'''
  138. Segment(H_1, K''') → l_3
  139. Midpoint(C, H_1) → I_1
  140. Rotate(H_1, If[15 < o ≤ 16, ((o - 15) * π), π], I_1) → J_1
  141. Semicircle(C, H_1) → p_3
  142. OrthogonalLine(J'_1, j) → m_3
  143. Intersect(p_3, m_3, 1) → K_1
  144. CircleSector(I_1, H_1, J_1) → q_3
  145. CircleArc(I_1, H_1, J_1) → r_3
  146. Angle(K_1, J'_1, C) → ο
  147. Dilate(K_1, If[17 < o ≤ 18, 18 - o, 0], J'_1) → L_1
  148. Segment(K_1, L_1) → n_3
  149. Line(J'_1, g_3) → s_3
  150. Intersect(q_3, r_3, 1) → M_1
  151. Intersect(r_3, s_3) → N_1, R_1
  152. Dilate(E_1, If[18 < o ≤ 19, 19 - o, 0], J'_1) → O_1
  153. Vector(E_1, O_1) → w_1
  154. Translate(L, w_1) → L'_1
  155. Translate(L', w_1) → L''
  156. Segment(L'_1, L'') → t_3
  157. Point(s_3) → P_1
  158. Angle(P_1, J'_1, K_1) → σ
  159. Rotate(L'_1, If[19 < o ≤ 20, ((o - 19) * σ), σ], J'_1) → Q_1
  160. Rotate(L'', If[19 < o ≤ 20, ((o - 19) * σ), σ], J'_1) → L'''
  161. Segment(Q_1, L''') → a_3

AI 绘图提示语参考

在一个直角三角形内部，绘制三个不同的小三角形，并展示它们内切圆之间的关系。首先画出直角三角形，然后找到直角顶点，以斜边为直径画一个半圆。接着，从圆与直角边交点向斜边作垂线，交点再分别向两条直角边作垂线，得到三个小三角形的顶点。分别画出这些小三角形及其内切圆，并标注关键角度，展示内切圆半径之间的几何平均关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。