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Medians and Equal Areas!

Medians and Equal Areas!
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2018-07-27 22:44
TriangleVectorGeometric Construction
This example demonstrates how medians of a triangle divide it into equal areas. Key concepts include the centroid of a triangle, vectors, and geometric construction. Using GeoGebra commands such as Segment, Midpoint, Intersect, etc., the properties of medians are visually illustrated. Suitable for middle school and above students to learn geometry and vector concepts.

GeoGebra Commands & Parameters

  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. Segment(D'_2, O) → g_1
  64. Polygon(A, H, F) → poly10, f_4, a_4, h_{10}
  65. Dilate(H, If[11 < m ≤ 12, 12 - m, 0], M) → H'_1
  66. Segment(A, F) → l_1
  67. Segment(H, H'_1) → m_1
  68. Dilate(F, If[12.5 < m ≤ 13.5, 13.5 - m, 0], C) → F'_1
  69. Vector(F, F'_1) → u
  70. Translate(A, u) → A'_1
  71. Translate(F, u) → F'_2
  72. Segment(A'_1, F'_2) → n_1
  73. Polygon(A'_1, H, F'_1) → poly11, f'_1, a'_1, h_{11}
  74. Dilate(F, If[14.5 < m ≤ 15.5, 15.5 - m, 0], K) → F'_3
  75. Segment(F, F'_3) → p_1
  76. Segment(C, H) → q_1
  77. Angle(F, K, H) → ζ
  78. Polygon(H, E, C) → poly12, c_4, h_{12}, e_4
  79. Line(C, D) → r_1
  80. OrthogonalLine(E, r_1) → s_1
  81. Intersect(s_1, r_1) → J
  82. Angle(E, J, C) → η
  83. Dilate(E, If[14.5 < m ≤ 15.5, 15.5 - m, 0], J) → E'
  84. Segment(E, E') → t_1
  85. Midpoint(K, J) → R
  86. Rotate(F, If[16 < m ≤ 17, ((m - 16) * π), 3.141592653589793], R) → F'_4
  87. Rotate(F'_3, If[16 < m ≤ 17, ((m - 16) * π), 3.141592653589793], R) → S
  88. Segment(F'_4, S) → g_2
  89. Dilate(H, If[17.5 < m ≤ 18.5, 18.5 - m, 0], N) → H'_2
  90. Segment(H, H'_2) → j_2
  91. Segment(C, E) → i_2
  92. Dilate(C, If[19 < m ≤ 20, 20 - m, 0], E) → C'_1
  93. Vector(C, C'_1) → w
  94. Translate(E, w) → E'_1
  95. Translate(C, w) → C'_2
  96. Translate(E, w) → E'_2
  97. Segment(C'_2, E'_2) → k_2
  98. Polygon(C'_1, H, E'_1) → poly13, e'_1, c'_1, h_{13}
  99. Line(F, B) → l_2
  100. OrthogonalLine(E, l_2) → m_2
  101. OrthogonalLine(D, l_2) → n_2
  102. Intersect(l_2, n_2) → T
  103. Intersect(m_2, l_2) → U
  104. Segment(H, B) → p_2
  105. Dilate(E, If[20.5 < m ≤ 21.5, 21.5 - m, 0], U) → E'_3
  106. Dilate(D, If[20.5 < m ≤ 21.5, 21.5 - m, 0], T) → D'_3
  107. Segment(E, E'_3) → q_2
  108. Segment(D, D'_3) → r_2
  109. Angle(E, U, F) → θ
  110. Angle(D, T, B) → ι
  111. Segment(H, B) → s_2
  112. Midpoint(U, T) → V
  113. Rotate(E, If[22 < m ≤ 23, ((m - 22) * π), 3.141592653589793], V) → E'_4
  114. Rotate(E'_3, If[22 < m ≤ 23, ((m - 22) * π), 3.141592653589793], V) → W
  115. Segment(E'_4, W) → t_2
  116. Polygon(B, D, H) → poly14, h_{14}, b_5, d_5
  117. Polygon(A, D, H) → poly15, h_{15}, a_5, d_6
  118. Polygon(A, H, F) → poly16, f_5, a_6, h_{16}
  119. Polygon(F, H, C) → poly17, c_5, f_6, h_{17}
  120. Polygon(H, E, C) → poly18, c_6, h_{18}, e_5