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Proof of Triangle Area (Related to Rectangle)

Proof of Triangle Area (Related to Rectangle)
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2017-06-12 19:59
TriangleFunctionCoordinate Geometry
This example demonstrates proving the area formula of a triangle by constructing a related rectangle. Key mathematical concepts include the triangle area formula, the rectangle area formula, and geometric transformations. Using GeoGebra commands such as Line, Point, Polygon, etc., the relationship between the triangle and the rectangle is visually illustrated. Suitable for middle school and above students to learn geometric proofs and area calculations.

GeoGebra Commands & Parameters

  1. Line(A, xAxis) → f_1
  2. Point(f_1) → M
  3. Ray(A, M) → h_1
  4. Point(h_1) → C
  5. Vector(C, A) → v
  6. Midpoint(B, C) → M_{BC}
  7. Rotate(A, (-ζ), M_{BC}) → I
  8. Segment(C, A) → i
  9. OrthogonalLine(I, i) → q
  10. Intersect(q, i) → G_1
  11. Vector(A, C) → u
  12. Midpoint(B, A) → M_{AB}
  13. Rotate(C, ε, M_{AB}) → J
  14. OrthogonalLine(J, i) → k
  15. Intersect(k, i) → E_1
  16. Polygon(A, B, C) → Area, c, a, b
  17. Angle(C, A, B) → α
  18. Angle(A, B, C) → β
  19. Angle(B, C, A) → γ
  20. Segment(A, B) → g
  21. Segment(B, C) → h
  22. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  23. OrthogonalLine(B, i) → j
  24. Intersect(j, i) → D
  25. Polygon(B, D, C) → poly2, c_2, b_2, d_1
  26. Polygon(B, A, D) → poly3, d_2, b_3, a_2
  27. Rotate(B, (-δ), M_{BC}) → B'
  28. Rotate(D, (-δ), M_{BC}) → L
  29. Rotate(C, (-δ), M_{BC}) → C'
  30. Polygon(B', L, C') → poly2', c', b', d'
  31. Rotate(B, δ, M_{AB}) → B'_1
  32. Rotate(A, δ, M_{AB}) → A'
  33. Rotate(D, δ, M_{AB}) → K
  34. Polygon(B'_1, A', K) → poly3', d'_1, b'_1, a'
  35. Segment(B, D) → d
  36. Rotate(A, ε, M_{AB}) → A'_1
  37. Rotate(B, ε, M_{AB}) → B'_2
  38. Polygon(A'_1, B'_2, J) → poly1', c'_1, a'_1, b'_2
  39. Polygon(Translate[J, Vector[(l * u)]], Translate[E_1, Vector[(l * u)]], Translate[A, Vector[(l * u)]]) → poly5, m, n, p
  40. Polygon(J, E_1, A) → poly4, a_3, c'_2, e_1
  41. Intersect(m, n) → F_1
  42. Rotate(B, (-ζ), M_{BC}) → B'_3
  43. Rotate(C, (-ζ), M_{BC}) → C'_2
  44. Polygon(I, B'_3, C'_2) → poly1'_1, c'_3, a'_2, b'_3
  45. Polygon(Translate[I, Vector[(o * v)]], Translate[G_1, Vector[(o * v)]], Translate[C, Vector[(o * v)]]) → poly7, r, s, t
  46. Polygon(I, G_1, C) → poly6, c_3, a'_3, g_1
  47. Intersect(r, s) → H_1
  48. Polygon(J, A, E_1) → poly8, e_2, c'_4, a_4
  49. Polygon(I, G_1, C) → poly9, c_4, a'_4, g_2
  50. Intersect(e_2, a_4) → I_1
  51. Intersect(e_2, c'_4) → J_1
  52. Point(a_4) → K_1
  53. Polygon(I_1, J_1, K_1) → poly10, k_1, i_1, j_1
  54. Mirror(C, A) → C'_1
  55. Mirror(A, C) → A'_2
  56. OrthogonalLine(C, b'_2) → l_1
  57. Intersect(l_1, b'_2) → E
  58. Polygon(E, C, B) → poly11, b_4, n_1, c_5
  59. Vector(B, J) → w
  60. Translate(E, w) → N'
  61. Translate(C, w) → C'_3
  62. Translate(B, w) → B'_4
  63. Polygon(Translate[E, Vector[(l * w)]], Translate[C, Vector[(l * w)]], Translate[B, Vector[(l * w)]]) → poly11', m_1, p_1, q_1
  64. Intersect(q_1, m_1) → F
  65. OrthogonalLine(A, b'_3) → r_1
  66. Intersect(b'_3, r_1) → G
  67. Polygon(B, G, A) → poly12, s_1, t_1, f_2
  68. Vector(B, I) → u_1
  69. Translate(B, u_1) → B'_5
  70. Translate(G, u_1) → H
  71. Translate(A, u_1) → A'_3
  72. Polygon(Translate[B, Vector[(o * u_1)]], Translate[G, Vector[(o * u_1)]], Translate[A, Vector[(o * u_1)]]) → poly12', h_2, i_2, j_2
  73. Intersect(h_2, i_2) → N