MathPlotAI Logo MathPlotAI

Parallelogram Theorem (3) Illustrated Without Words

Parallelogram Theorem (3) Illustrated Without Words
Loading the math board and drawing, please wait…
2016-10-16 09:03
PolygonAngleVectorTransformationGeometric Construction
In collections Congruence (CO) Cluster 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 Unit 7: Polygons and Quadrilaterals Quadrilaterals
This GeoGebra example demonstrates a parallelogram theorem (Theorem 3) through a purely visual, wordless illustration. The core mathematical concepts involve properties of parallelograms, diagonals, congruent triangles, and relationships between segments or areas. The construction uses commands such as Rotate, Angle, Ray, Line, Point, Intersect, Segment, Min, Circle, Dilate, CircleSector, Vector, Translate, Polygon, and Midpoint, creating multiple objects (points A, B, C, D; segments a, b, c; circle c_1, etc.) to reveal the theorem dynamically. Suitable for middle or high school geometry classes, this example helps students intuitively grasp parallelogram theorems and can be used for exploratory learning or teacher demonstrations.

GeoGebra Commands & Parameters

  1. Rotate(B, a°, A) → B'
  2. Angle(B, A, B') → α
  3. Ray(A, B) → f
  4. Ray(A, B') → g
  5. Line(B, g) → h
  6. Point(g) → C
  7. Line(C, f) → i
  8. Intersect(h, i) → D
  9. Segment(C, D) → j
  10. Segment(A, B) → k
  11. Segment(B, D) → l
  12. Segment(A, C) → m
  13. Min(m, k) → b
  14. Circle(C, (0.2 * b)) → d
  15. Circle(A, (0.2 * b)) → e
  16. Circle(B, (0.2 * b)) → p
  17. Circle(D, (0.2 * b)) → c
  18. Intersect(e, k, 1) → E
  19. Intersect(e, m, 1) → F
  20. Intersect(p, k, 1) → G
  21. Intersect(p, l, 1) → H
  22. Intersect(d, g, 2) → I
  23. Ray(C, I) → q
  24. Intersect(c, h, 2) → J
  25. Ray(D, J) → r
  26. Intersect(c, i, 2) → K
  27. Ray(D, K) → s
  28. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], B) → A'
  29. Segment(A, A') → t
  30. Rotate(E, If[1 < o ≤ 2, ((o - 1) * α), α], A) → E'
  31. CircleSector(A, E, E') → c_1
  32. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], C) → A'_1
  33. Vector(A, A'_1) → u
  34. Translate(c_1, u) → d_1
  35. Segment(A, A'_1) → f_1
  36. Dilate(C, If[3 < o ≤ 4, 4 - o, 0], D) → C'
  37. Vector(C, C') → v
  38. Translate(d_1, v) → e_1
  39. Segment(C, C') → g_1
  40. Dilate(D, If[4 < o ≤ 5, 5 - o, 0], B) → D'
  41. Segment(D, D') → h_1
  42. Polygon(A, B, D, C) → poly1, a_1, b_1, d_2, c_2
  43. Dilate(A, If[5.5 < o ≤ 6.5, 6.5 - o, 0], C) → A'_2
  44. Vector(A, A'_2) → w
  45. Translate(A, w) → A'_3
  46. Translate(A', w) → A''
  47. Dilate(A, If[6.5 < o ≤ 7.5, 7.5 - o, 0], B) → A'_4
  48. Vector(A, A'_4) → u_1
  49. Translate(A, u_1) → A'_5
  50. Translate(A'_1, u_1) → L
  51. Intersect(d, i) → M, N
  52. Intersect(d, g) → O, P
  53. Angle(A, C, N) → β
  54. Rotate(N, If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * β), (-β)], C) → N'
  55. Dilate(C, If[7.5 < o ≤ 8.5, 8.5 - o, 0], B) → C'_1
  56. Vector(C, C'_1) → v_1
  57. Midpoint(A, D) → Q
  58. Dilate(A, If[5.5 < o ≤ 6.5, 6.5 - o, 0], Q) → A'_6
  59. Segment(A, A'_6) → i_1
  60. Dilate(Q, If[6.5 < o ≤ 7.5, 7.5 - o, 0], D) → Q'
  61. Segment(Q, Q') → j_1
  62. Dilate(B, If[7.5 < o ≤ 8.5, 8.5 - o, 0], Q) → B'_1
  63. Segment(B, B'_1) → k_1
  64. Dilate(Q, If[8.5 < o ≤ 9.5, 9.5 - o, 0], C) → Q'_1
  65. Segment(Q, Q'_1) → l_1
  66. Rotate(A, If[9.5 < o ≤ 10.5, ((-(o - 9.5)) * π), (-π)], Q) → A'_7
  67. Rotate(A'_6, If[9.5 < o ≤ 10.5, ((-(o - 9.5)) * π), (-π)], Q) → R
  68. Segment(A'_7, R) → m_1
  69. Rotate(B, If[10.5 < o ≤ 11.5, ((-(o - 10.5)) * π), (-π)], Q) → B'_2
  70. Rotate(B'_1, If[10.5 < o ≤ 11.5, ((-(o - 10.5)) * π), (-π)], Q) → S
  71. Segment(B'_2, S) → n_1

AI Drawing Prompt Reference

Draw a parallelogram ABCD, connect its diagonals AC and BD, construct points E, F, G, H, I, J, K, L such that AE = AF = AG = AH = AI = AJ = AK = AL, then construct circles with these points as centers and radius equal to AE, and finally use translation and rotation to form a series of parallelograms, illustrating the parallelogram theorem.
The prompt above was compiled from the original drawing commands and is for reference only.