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三角形中线与等面积现象

三角形中线与等面积现象
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2016-05-14 23:55
测试改名0多边形角变换几何作图
所属专题 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Geometry Review Triangles: Points of Concurrency
该范例演示了三角形三条中线交于重心，并将三角形分割成六个面积相等的小三角形这一经典几何现象。核心数学概念包括三角形的中线、重心（形心）以及面积等分性质。作图要点：使用GeoGebra的Segment（线段）、Midpoint（中点）、Intersect（交点）等命令构造三角形及其中线，通过Polygon（多边形）命令生成分割后的小三角形，并利用Dilate（缩放）、Translate（平移）、Rotate（旋转）等变换工具辅助观察面积关系。适合中学几何教学，用于直观理解重心性质与面积守恒，也可作为微积分中积分与面积概念的直观铺垫。

GeoGebra 命令及参数

  1. Segment(A, B) → a
  2. Segment(C, A) → b
  3. Segment(C, B) → c
  4. Midpoint(a) → D
  5. Midpoint(c) → E
  6. Midpoint(b) → F
  7. Segment(C, D) → d
  8. Segment(A, E) → e
  9. Segment(B, F) → f
  10. Intersect(f, e) → H
  11. OrthogonalLine(D, e) → g
  12. OrthogonalLine(F, e) → h
  13. OrthogonalLine(E, d) → i
  14. OrthogonalLine(F, d) → j
  15. Intersect(g, e) → G
  16. Intersect(h, e) → I
  17. Intersect(j, d) → K
  18. OrthogonalLine(E, f) → k
  19. OrthogonalLine(D, f) → l
  20. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  21. Dilate(A, If[1 < m ≤ 2, 2 - m, 0], E) → A'
  22. Dilate(C, If[0 < m ≤ 1, 1 - m, 0], D) → C'
  23. Dilate(B, If[2 < m ≤ 3, 3 - m, 0], F) → B'
  24. Segment(A, A') → n
  25. Segment(B, B') → p
  26. Segment(C, C') → q
  27. OrthogonalLine(H, a) → r
  28. OrthogonalLine(H, b) → s
  29. OrthogonalLine(H, c) → t
  30. Intersect(r, a) → L
  31. Intersect(s, b) → M
  32. Intersect(t, c) → N
  33. Dilate(B, If[4 < m ≤ 5, 5 - m, 0], D) → B'_1
  34. Dilate(H, If[4 < m ≤ 5, 5 - m, 0], L) → H'
  35. Segment(H, H') → e_1
  36. Angle(H, L, A) → α
  37. Angle(H, M, C) → β
  38. Angle(H, N, B) → γ
  39. Segment(B, D) → d_1
  40. Segment(H, H') → h_1
  41. Polygon(D, B, H) → poly2, h_2, d_2, b_2
  42. Dilate(B, If[6 < m ≤ 7, 7 - m, 0], D) → B'_3
  43. Vector(B, B'_3) → v
  44. Translate(D, v) → D'
  45. Segment(B'_3, D') → i_1
  46. Polygon(D', H, B'_3) → poly3, b'_3, d', h_3
  47. Midpoint(I, G) → P
  48. Dilate(D, If[8 < m ≤ 9, 9 - m, 0], G) → D'_1
  49. Segment(D, D'_1) → j_1
  50. Angle(D, G, H) → δ
  51. Angle(F, I, A) → ε
  52. Dilate(F, If[8 < m ≤ 9, 9 - m, 0], I) → F'
  53. Segment(F, F') → k_1
  54. Polygon(D, B, H) → poly4, h_4, d_3, b_3
  55. Polygon(H, D, A) → poly5, a_2, h_5, d_4
  56. Polygon(A, H, F) → poly6, f_2, a_3, h_6
  57. Polygon(F, H, C) → poly7, c_2, f_3, h_7
  58. Polygon(H, E, C) → poly8, c_3, h_8, e_2
  59. Polygon(H, E, B) → poly9, b_4, h_9, e_3
  60. Segment(A, H) → f_1
  61. Rotate(D, If[9.5 < m ≤ 10.5, ((m - 9.5) * π), 3.141592653589793], P) → D'_2
  62. Rotate(D'_1, If[9.5 < m ≤ 10.5, ((m - 9.5) * π), 3.141592653589793], P) → O
  63. Polygon(A, H, F) → poly10, f_4, a_4, h_{10}
  64. Dilate(H, If[11 < m ≤ 12, 12 - m, 0], M) → H'_1
  65. Segment(A, F) → l_1
  66. Segment(H, H'_1) → m_1
  67. Dilate(F, If[12.5 < m ≤ 13.5, 13.5 - m, 0], C) → F'_1
  68. Vector(F, F'_1) → u
  69. Translate(A, u) → A'_1
  70. Translate(F, u) → F'_2
  71. Segment(A'_1, F'_2) → n_1
  72. Polygon(A'_1, H, F'_1) → poly11, f'_1, a'_1, h_{11}
  73. Dilate(F, If[14.5 < m ≤ 15.5, 15.5 - m, 0], K) → F'_3
  74. Segment(F, F'_3) → p_1
  75. Segment(C, H) → q_1
  76. Angle(F, K, H) → ζ
  77. Polygon(H, E, C) → poly12, c_4, h_{12}, e_4
  78. Line(C, D) → r_1
  79. OrthogonalLine(E, r_1) → s_1
  80. Intersect(s_1, r_1) → J
  81. Angle(E, J, C) → η
  82. Dilate(E, If[14.5 < m ≤ 15.5, 15.5 - m, 0], J) → E'
  83. Segment(E, E') → t_1
  84. Midpoint(K, J) → R
  85. Rotate(F, If[16 < m ≤ 17, ((m - 16) * π), 3.141592653589793], R) → F'_4
  86. Rotate(F'_3, If[16 < m ≤ 17, ((m - 16) * π), 3.141592653589793], R) → S
  87. Dilate(H, If[17.5 < m ≤ 18.5, 18.5 - m, 0], N) → H'_2
  88. Segment(H, H'_2) → j_2
  89. Segment(C, E) → i_2
  90. Dilate(C, If[19 < m ≤ 20, 20 - m, 0], E) → C'_1
  91. Vector(C, C'_1) → w
  92. Translate(E, w) → E'_1
  93. Translate(C, w) → C'_2
  94. Translate(E, w) → E'_2
  95. Segment(C'_2, E'_2) → k_2
  96. Polygon(C'_1, H, E'_1) → poly13, e'_1, c'_1, h_{13}
  97. Line(F, B) → l_2
  98. OrthogonalLine(E, l_2) → m_2
  99. OrthogonalLine(D, l_2) → n_2
  100. Intersect(l_2, n_2) → T
  101. Intersect(m_2, l_2) → U
  102. Segment(H, B) → p_2
  103. Dilate(E, If[20.5 < m ≤ 21.5, 21.5 - m, 0], U) → E'_3
  104. Dilate(D, If[20.5 < m ≤ 21.5, 21.5 - m, 0], T) → D'_3
  105. Segment(E, E'_3) → q_2
  106. Segment(D, D'_3) → r_2
  107. Angle(E, U, F) → θ
  108. Angle(D, T, B) → ι
  109. Segment(H, B) → s_2
  110. Midpoint(U, T) → V
  111. Rotate(E, If[22 < m ≤ 23, ((m - 22) * π), 3.141592653589793], V) → E'_4
  112. Rotate(E'_3, If[22 < m ≤ 23, ((m - 22) * π), 3.141592653589793], V) → W
  113. Polygon(B, D, H) → poly14, h_{14}, b_5, d_5
  114. Polygon(A, D, H) → poly15, h_{15}, a_5, d_6
  115. Polygon(A, H, F) → poly16, f_5, a_6, h_{16}
  116. Polygon(F, H, C) → poly17, c_5, f_6, h_{17}
  117. Polygon(H, E, C) → poly18, c_6, h_{18}, e_5
  118. Dilate(D, If[9.5 < m ≤ 10.5, 10.5 - m, 0], F') → C'_3
  119. Dilate(D, If[9.5 < d ≤ 10.5, 10.5 - d, 0], F') → D'_4
  120. Segment(C'_3, D'_4) → d'_1
  121. Vector(D, C'_3) → u_1
  122. Translate(D, u_1) → D'_5
  123. Translate(D'_1, u_1) → Q
  124. Segment(D'_5, Q) → g_3
  125. Dilate(F, If[16 < m ≤ 17, 17 - m, 0], E') → F'_5
  126. Vector(F, F'_5) → v_1
  127. Translate(F, v_1) → F'_6
  128. Translate(F'_3, v_1) → Z
  129. Segment(F'_6, Z) → g_1
  130. Dilate(E, If[22 < m ≤ 23, 23 - m, 0], D'_3) → D'_6
  131. Vector(E, D'_6) → w_1
  132. Translate(E, w_1) → E'_5
  133. Translate(E'_3, w_1) → A_1
  134. Segment(E'_5, A_1) → g_2