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Special Quadrilateral: Theorem 4

Special Quadrilateral: Theorem 4
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2017-02-10 06:42
PolygonAngleTransformation
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 Unit 7: Polygons and Quadrilaterals Quadrilaterals
This example demonstrates Theorem 4 of special quadrilaterals, involving parallelograms and similar triangles. It is constructed using GeoGebra commands such as rotation, angle, line, and segment, suitable for middle school and above students to learn geometric construction.

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(B, If[0 < o ≤ 1, 1 - o, 0], A) → A'
  29. Rotate(E, If[1 < o ≤ 2, ((o - 1) * α), α], A) → E'
  30. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], C) → A'_1
  31. Vector(A, A'_1) → u
  32. Segment(A, A'_1) → f_1
  33. Dilate(C, If[2 < o ≤ 3, 3 - o, 0], D) → C'
  34. Vector(C, C') → v
  35. Dilate(B, If[3 < o ≤ 4, 4 - o, 0], D) → D'
  36. Polygon(A, B, D, C) → poly1, a_1, b_1, d_2, c_2
  37. Dilate(A, If[5.5 < o ≤ 6.5, 6.5 - o, 0], C) → A'_2
  38. Vector(A, A'_2) → w
  39. Translate(A, w) → A'_3
  40. Translate(A', w) → A''
  41. Dilate(A, If[6.5 < o ≤ 7.5, 7.5 - o, 0], B) → A'_4
  42. Vector(A, A'_4) → u_1
  43. Translate(A, u_1) → A'_5
  44. Translate(A'_1, u_1) → L
  45. Intersect(d, i) → M, N
  46. Intersect(d, g) → O, P
  47. Angle(A, C, N) → β
  48. Rotate(N, If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * β), (-β)], C) → N'
  49. Dilate(C, If[7.5 < o ≤ 8.5, 8.5 - o, 0], B) → C'_1
  50. Vector(C, C'_1) → v_1
  51. Line(A, B) → p_1
  52. Intersect(e, p_1) → T, U
  53. Ray(A, T) → q_1
  54. Intersect(e, f_1) → V,
  55. Intersect(d, i, 1) → W
  56. Ray(C, W) → r_1
  57. Segment(B, A') → t
  58. Rotate(V, If[2 < o ≤ 3, ((o - 2) * (π - α)), π - α], A) → V'
  59. CircleSector(A, V, V') → c_1
  60. Dilate(A, If[12 < o ≤ 13, 13 - o, 0], C) → A'_8
  61. Vector(A, A'_8) → w_1
  62. Dilate(C, If[13 < o ≤ 14, 14 - o, 0], D) → C'_2
  63. Vector(C, C'_2) → e_1
  64. Dilate(A, If[3 < o ≤ 4, 4 - o, 0], C) → A'_9
  65. Vector(A, A'_9) → g_1
  66. Translate(B, g_1) → B'_3
  67. Translate(A', g_1) → A''_1
  68. Segment(B'_3, A''_1) → t'
  69. Segment(B, D') → h_1
  70. Dilate(A, If[4 < o ≤ 5, 5 - o, 0], C) → A'_6
  71. Vector(A, A'_6) → d_1
  72. Translate(c_1, d_1) → k_1
  73. Dilate(C, If[6 < o ≤ 7, 7 - o, 0], D) → C'_3
  74. Vector(C, C'_3) → i_1
  75. Translate(k_1, i_1) → s_1

AI Drawing Prompt Reference

Draw a parallelogram ABCD with vertices A, B, C, and D. First, draw the sides AB, BC, CD, and DA. Then, use rotation and angles to construct similar triangles. Next, draw the diagonals AC and BD. After that, use rotation and scaling to draw circles. Finally, use these circles and line segments to construct the symmetric figure of the parallelogram.
The prompt above was compiled from the original drawing commands and is for reference only.