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Special Quadrilateral: Theorem 2

Special Quadrilateral: Theorem 2
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2017-02-09 23:25
PolygonAngleTransformationGeometric 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 Unit 7: Polygons and Quadrilaterals Quadrilaterals
This example demonstrates Theorem 2 of special quadrilaterals, involving parallelograms and related properties. It is dynamically constructed using commands like Rotate, Angle, Ray, Line, and Segment, suitable for middle school and above students.

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. Segment(A, A'_1) → f_1
  35. Dilate(C, If[3 < o ≤ 4, 4 - o, 0], D) → C'
  36. Vector(C, C') → v
  37. Segment(C, C') → g_1
  38. Dilate(D, If[4 < o ≤ 5, 5 - o, 0], B) → D'
  39. Segment(D, D') → h_1
  40. Polygon(A, B, D, C) → poly1, a_1, b_1, d_2, c_2
  41. Dilate(A, If[5.5 < o ≤ 6.5, 6.5 - o, 0], C) → A'_2
  42. Vector(A, A'_2) → w
  43. Translate(A, w) → A'_3
  44. Translate(A', w) → A''
  45. Dilate(A, If[6.5 < o ≤ 7.5, 7.5 - o, 0], B) → A'_4
  46. Vector(A, A'_4) → u_1
  47. Translate(A, u_1) → A'_5
  48. Translate(A'_1, u_1) → L
  49. Intersect(d, i) → M, N
  50. Intersect(d, g) → O, P
  51. Angle(A, C, N) → β
  52. Rotate(N, If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * β), (-β)], C) → N'
  53. Dilate(C, If[7.5 < o ≤ 8.5, 8.5 - o, 0], B) → C'_1
  54. Vector(C, C'_1) → v_1
  55. Intersect(d, f_1) → Q,
  56. Intersect(d, g_1) → R,
  57. Rotate(R, If[5 < o ≤ 6, ((-(o - 5)) * (π - α)), (-(π - α))], C) → R'
  58. CircleSector(C, R', R) → d_1
  59. Dilate(A, If[6 < o ≤ 7, 7 - o, 0], D) → A'_6
  60. Vector(A, A'_6) → w_1
  61. Translate(c_1, w_1) → e_1
  62. Rotate(e_1, If[7 < o ≤ 8, ((-(o - 7)) * π), (-π)], D) → k_1
  63. Dilate(C, If[8 < o ≤ 9, 9 - o, 0], B) → C'_2
  64. Vector(C, C'_2) → i_1
  65. Translate(d_1, i_1) → p_1
  66. Rotate(p_1, If[9 < o ≤ 10, ((-(o - 9)) * π), (-π)], B) → q_1
  67. Dilate(A, If[11 < o ≤ 12, 12 - o, 0], C) → A'_7
  68. Vector(A, A'_7) → j_1
  69. Translate(c_1, j_1) → r_1
  70. Dilate(C, If[12 < o ≤ 13, 13 - o, 0], D) → C'_3
  71. Vector(C, C'_3) → l_1
  72. Translate(r_1, l_1) → s_1

AI Drawing Prompt Reference

Draw a parallelogram ABCD with points A, B, C, and D connected in sequence. First, rotate point B around point A by α degrees to get point B'. Then, construct ray g through points A and B'. Next, construct line h through point B and ray g, and point C is the intersection of line h and ray g. After that, construct line i through points C and line f, and point D is the intersection of line i and line h. Finally, connect points A, B, C, and D to form parallelogram ABCD.
The prompt above was compiled from the original drawing commands and is for reference only.