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Isosceles Trapezoid Midpoints Form a Rhombus: Coordinate Geometry Proof

Isosceles Trapezoid Midpoints Form a Rhombus: Coordinate Geometry Proof
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2018-12-04 10:07
PolygonVectorCoordinate GeometryTransformationGeometric Construction
This GeoGebra example demonstrates that the midpoints of the four sides of an isosceles trapezoid, when connected in order, form a rhombus. It serves as a coordinate geometry proof prompt. Core concepts include properties of isosceles trapezoids (equal legs, base angles, symmetry), midpoint coordinates, rhombus criteria (equal side lengths or perpendicular bisecting diagonals), and distance formulas in the Cartesian plane. The construction uses GeoGebra commands such as Line, Ray, Point, Midpoint, Slope, OrthogonalLine, Mirror, Dilate, Segment, Circle, Intersect, Vector, Translate, Angle, Rotate, and Polygon to build the trapezoid, locate midpoints, and verify the resulting quadrilateral. Suitable for high school geometry or coordinate geometry classes, this interactive applet helps students visualize the theorem and develop algebraic proof skills.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Ray(A, B) → b
  3. Point(b) → C
  4. Midpoint(A, C) → D
  5. Line(A, C) → c
  6. Slope(c) → d
  7. OrthogonalLine(D, c) → g
  8. Point(g) → E
  9. Ray(D, E) → h
  10. Point(h) → F
  11. Slope(h) → i
  12. Line(F, c) → l
  13. Point(l) → G
  14. Point(l) → H
  15. Ray(F, G) → m
  16. Ray(F, H) → n
  17. Point(m) → I
  18. Mirror(I, h) → I'
  19. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], C) → A'
  20. Segment(A, A') → p
  21. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], I) → A'_1
  22. Segment(A, A'_1) → q
  23. Segment(A, C) → r
  24. Segment(C, I') → s
  25. Segment(I', I) → t
  26. Segment(I, A) → a_1
  27. Circle(A, (0.2 * Min[a_1, r])) → c_1
  28. Circle(I, (0.2 * Min[a_1, r])) → d_1
  29. Intersect(d_1, a_1, 1) → J
  30. Intersect(d_1, t, 1) → K
  31. Intersect(c_1, r, 1) → L
  32. Intersect(c_1, a_1, 1) → M
  33. Dilate(I, If[2 < o ≤ 3, 3 - o, 0], I') → I'_1
  34. Segment(I, I'_1) → b_1
  35. Dilate(A, If[3 < o ≤ 4, 4 - o, 0], C) → A'_2
  36. Vector(A, A'_2) → u
  37. Translate(A, u) → A'_3
  38. Translate(A'_1, u) → N
  39. Segment(A'_3, N) → q'
  40. OrthogonalLine(A, r) → g_1
  41. Point(g_1) → O
  42. Angle(I, A, O) → α
  43. Angle(I', C, A) → β
  44. Rotate(A'_3, If[4 < o ≤ 5, (((o - 4) * 2) * α), (2 * α)], C) → P
  45. Rotate(N, If[4 < o ≤ 5, (((o - 4) * 2) * α), (2 * α)], C) → N'
  46. Segment(P, N') → q''
  47. Dilate(A, If[5 < o ≤ 6, 6 - o, 0], C) → A'_4
  48. Vector(A, A'_4) → v
  49. Dilate(I, If[7 < o ≤ 8, 8 - o, 0], I') → I'_2
  50. Vector(I, I'_2) → w
  51. Dilate(A, If[5 < o ≤ 6, 6 - o, 0], I') → A'_5
  52. Dilate(A, If[6 < o ≤ 7, 7 - o, 0], I) → A'_6
  53. Vector(A, A'_6) → u_1
  54. Translate(A, u_1) → A'_7
  55. Translate(A'_5, u_1) → Q
  56. Segment(I, C) → l_1
  57. Angle(L, A, I') → γ
  58. Rotate(A'_7, If[11 < o ≤ 12, ((-(o - 11)) * (2 * γ)), (-2 * γ)], I) → R
  59. Rotate(Q, If[7 < o ≤ 8, (((-(o - 7)) * 2) * γ), (-2 * γ)], I) → Q'
  60. Polygon(A, C, I', I) → poly1, a_2, c_2, i', i_2
  61. Midpoint(a_1) → S
  62. Midpoint(t) → T
  63. Midpoint(s) → U
  64. Midpoint(r) → V
  65. Dilate(A, If[5.5 < o ≤ 6.5, 6.5 - o, 0], S) → A'_8
  66. Dilate(I, If[5.5 < o ≤ 6.5, 6.5 - o, 0], T) → I'_3
  67. Dilate(I', If[5.5 < o ≤ 6.5, 6.5 - o, 0], U) → I''
  68. Dilate(C, If[5.5 < o ≤ 6.5, 6.5 - o, 0], V) → C'
  69. Segment(A, A'_8) → f_1
  70. Segment(I, I'_3) → h_1
  71. Segment(I', I'') → i_1
  72. Segment(C, C') → j_1
  73. Rotate(A, If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * π), (-π)], S) → A'_9
  74. Rotate(A'_8, If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * π), (-π)], S) → W
  75. Segment(A'_9, W) → k_1
  76. Rotate(I, If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * π), (-π)], T) → I'_4
  77. Rotate(I'_3, If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * π), (-π)], T) → Z
  78. Segment(I'_4, Z) → m_1
  79. Rotate(I', If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * π), (-π)], U) → I''_1
  80. Rotate(I'', If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * π), (-π)], U) → I'''
  81. Segment(I''_1, I''') → n_1
  82. Rotate(C, If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * π), (-π)], V) → C'_1
  83. Rotate(C', If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * π), (-π)], V) → C''
  84. Segment(C'_1, C'') → p_1
  85. Dilate(V, If[7.5 < o ≤ 8.5, 8.5 - o, 0], S) → V'
  86. Segment(V, V') → q_1
  87. Angle(V, S, T) → δ
  88. Rotate(V, If[8.5 < o ≤ 9.5, ((o - 8.5) * δ), δ], S) → V'_1
  89. Rotate(V', If[8.5 < o ≤ 9.5, ((o - 8.5) * δ), δ], S) → V''
  90. Segment(V'_1, V'') → r_1
  91. Rotate(V'_1, If[9.5 < o ≤ 10.5, ((o - 9.5) * (π - δ)), π - δ], T) → A_1
  92. Rotate(V'', If[9.5 < o ≤ 10.5, ((o - 9.5) * (π - δ)), π - δ], T) → V'''
  93. Segment(A_1, V''') → s_1
  94. Rotate(A_1, If[10.5 < o ≤ 11.5, ((o - 10.5) * δ), δ], U) → B_1
  95. Rotate(V''', If[10.5 < o ≤ 11.5, ((o - 10.5) * δ), δ], U) → C_1
  96. Segment(B_1, C_1) → t_1
  97. Polygon(A, V, S) → poly2, s_2, a_3, v_1
  98. Polygon(S, I, T) → poly3, t_2, s_3, i_3
  99. Polygon(T, U, I') → poly4, i'_1, t_3, u_2
  100. Polygon(V, C, U) → poly5, u_3, v_2, c_3
  101. Polygon(V, U, T, S) → poly6, v_3, u_4, t_4, s_4

AI Drawing Prompt Reference

Draw an isosceles trapezoid ABCD, then find the midpoints E and F of sides AB and CD, respectively. Connect E and F to form a quadrilateral EFGH. Verify that EFGH is a rhombus.
The prompt above was compiled from the original drawing commands and is for reference only.