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Proof Without Words: Parallelogram Properties

Proof Without Words: Parallelogram Properties
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2016-01-14 17:44
PolygonVectorGeometric Construction
This example demonstrates the properties of a parallelogram by dragging points and sliders. It proves that the diagonals of a parallelogram bisect each other. Using GeoGebra commands such as points, line segments, vectors, and translations, the construction visually illustrates this geometric property. Suitable for middle school and above students to learn about geometric transformations and proofs.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Intersect(xAxis, yAxis) → B
  3. CircleArc(B, D, A) → c
  4. Point(c) → E
  5. Segment(A, E) → a
  6. Segment(E, F) → b
  7. Segment(C, E) → d
  8. Segment(E, G) → e
  9. Line(F, a) → f
  10. Line(G, d) → g
  11. Intersect(f, g) → H
  12. Segment(A, C) → h
  13. Line(H, E) → i
  14. Intersect(i, h) → I
  15. Line(A, i) → j
  16. Line(C, i) → k
  17. Vector(H, E) → u
  18. Translate(I, u) → I'
  19. Line(I', h) → l
  20. Intersect(j, l) → J
  21. Intersect(k, l) → K
  22. Vector(E, A) → v
  23. Vector(E, C) → w
  24. Translate(G, w) → G'
  25. Translate(F, v) → F'
  26. Polygon(A, E, F, F') → poly1, a_1, e_1, f_1, f'
  27. Polygon(A, J, K, C) → poly2, a_2, j_1, k_1, c_1
  28. Polygon(E, G, G', C) → poly3, e_2, g_1, g', c_2
  29. Dilate(F, If[m < 1, 1 - m, 0], H) → F'_1
  30. Dilate(G, If[m < 1, 1 - m, 0], H) → G'_1
  31. Translate(F'_1, v) → L
  32. Translate(G'_1, w) → M
  33. Polygon(A, L, F'_1, E) → poly4, a_3, l_1, f'_1, e_3
  34. Polygon(E, G'_1, M, C) → poly5, e_4, g'_1, m_1, c_3
  35. Vector(E, I) → n
  36. Translate(H, n) → H'
  37. Dilate(H, If[1 ≤ m < 2, 2 - m, 0], H') → H'_1
  38. Vector(L, A) → p
  39. Translate(H'_1, p) → N
  40. Polygon(L, H'_1, N, A) → poly6, l_2, h'_1, n_1, a_4
  41. Polygon(H'_1, M, C, N) → poly7, h'_2, m_2, c_4, n_2
  42. Dilate(I, If[2 ≤ m < 3, 3 - m, 0], I') → I'_1
  43. Vector(A, L) → q
  44. Translate(I'_1, q) → O
  45. Vector(I, A) → r
  46. Translate(O, r) → O'
  47. Translate(I'_1, r) → P
  48. Vector(I, C) → s
  49. Translate(O, s) → O'_1
  50. Translate(I'_1, s) → Q
  51. Polygon(O, O', P, I'_1) → poly8, o, o', p_1, i'_1
  52. Polygon(O, O'_1, Q, I'_1) → poly9, o_1, o'_1, q_1, i'_2