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Midpoints of a Quadrilateral Form a Parallelogram

Midpoints of a Quadrilateral Form a Parallelogram
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2018-12-04 10:08
PolygonVectorGeometric Construction
This example dynamically illustrates that the midpoints of any quadrilateral always form a parallelogram. Key concepts include quadrilaterals, parallelograms, midpoints, and vectors. Using GeoGebra commands such as Polygon, Midpoint, and Dilate, this construction provides a visual proof of this geometric property. Suitable for middle school and above students studying geometric transformations and proofs.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → poly1, a, b, c, d
  2. Midpoint(A, B) → E
  3. Midpoint(B, C) → F
  4. Midpoint(C, D) → G
  5. Midpoint(D, A) → H
  6. Polygon(E, F, G, H) → poly2, e, f, g, h
  7. Dilate(A, If[0 < i ≤ 1, 1 - i, 0], E) → A'
  8. Dilate(B, If[0 < i ≤ 1, 1 - i, 0], F) → B'
  9. Dilate(C, If[0 < i ≤ 1, 1 - i, 0], G) → C'
  10. Dilate(D, If[0 < i ≤ 1, 1 - i, 0], H) → D'
  11. Segment(A, A') → j
  12. Segment(B, B') → k
  13. Segment(C, C') → l
  14. Segment(D, D') → m
  15. Dilate(E, If[1 < i ≤ 2, 2 - i, 0], B) → E'
  16. Dilate(F, If[1 < i ≤ 2, 2 - i, 0], C) → F'
  17. Dilate(G, If[1 < i ≤ 2, 2 - i, 0], D) → G'
  18. Dilate(H, If[1 < i ≤ 2, 2 - i, 0], A) → H'
  19. Segment(E, E') → n
  20. Segment(F, F') → p
  21. Segment(G, G') → q
  22. Segment(H, H') → r
  23. Rotate(A, If[2 < i ≤ 3, ((-(i - 2)) * π), (-π)], E) → A'_1
  24. Rotate(A', If[2 < i ≤ 3, ((-(i - 2)) * π), (-π)], E) → A''
  25. Segment(A'_1, A'') → j'
  26. Rotate(B, If[2 < i ≤ 3, ((-(i - 2)) * π), (-π)], F) → B'_1
  27. Rotate(B', If[2 < i ≤ 3, ((-(i - 2)) * π), (-π)], F) → B''
  28. Segment(B'_1, B'') → k'
  29. Rotate(C, If[2 < i ≤ 3, ((-(i - 2)) * π), (-π)], G) → C'_1
  30. Rotate(C', If[2 < i ≤ 3, ((-(i - 2)) * π), (-π)], G) → C''
  31. Segment(C'_1, C'') → l'
  32. Rotate(D, If[2 < i ≤ 3, ((-(i - 2)) * π), (-π)], H) → D'_1
  33. Rotate(D', If[2 < i ≤ 3, ((-(i - 2)) * π), (-π)], H) → D''
  34. Segment(D'_1, D'') → m'
  35. Dilate(E, If[3 < i ≤ 4, 4 - i, 0], F) → E'_1
  36. Segment(E, E'_1) → s
  37. Dilate(F, If[4 < i ≤ 5, 5 - i, 0], G) → F'_1
  38. Segment(F, F'_1) → t
  39. Dilate(E, If[5 < i ≤ 6, 6 - i, 0], H) → E'_2
  40. Vector(E, E'_2) → u
  41. Translate(E, u) → E'_3
  42. Translate(E'_1, u) → I
  43. Segment(E'_3, I) → s'
  44. Dilate(G, If[6 < i ≤ 7, 7 - i, 0], H) → G'_1
  45. Vector(G, G'_1) → v
  46. Translate(F, v) → F'_2
  47. Translate(F'_1, v) → J
  48. Segment(F'_2, J) → t'
  49. Polygon(E, F, G, H) → poly3, e_1, f_1, g_1, h_1