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Midpoints of a Quadrilateral Form a Parallelogram by Tim Brzezinski

Midpoints of a Quadrilateral Form a Parallelogram by Tim Brzezinski
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2016-06-18 05:41
PolygonVectorGeometric Construction
In collections Congruence (CO) Cluster Congruence (Vol 2) CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Gail LeBlancCCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Quadrilaterals
This example dynamically illustrates that the midpoints of any quadrilateral always form a parallelogram. Key mathematical concepts include quadrilaterals, parallelograms, and midpoints. GeoGebra commands used include Polygon, Midpoint, Dilate, etc. Suitable for middle school and above students to learn geometric construction 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