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Perigal's Dissection

Perigal's Dissection
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2015-04-15 20:10
Coordinate GeometryOptimizationGeometric Construction
In collections GCSE Higher Ch5 Angles & Trigonometry
This example demonstrates Perigal's Dissection, a geometric puzzle where two semicircles are dissected into pieces that can be rearranged to form a square. It involves coordinate geometry, subtraction, geometric construction, and special points. Key GeoGebra commands and objects used include Point, Semicircle, Polygon, Angle, Midpoint, Line, OrthogonalLine, Intersect, If, and Translate. Suitable for high school and above, exploring geometric transformations and optimization problems.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Semicircle(A, B) → c
  4. Point(c) → C
  5. Polygon(A, C, 4) → poly1, a, b, d, e, D, E
  6. Polygon(C, B, 4) → poly2, f, g, h, i, F, G
  7. Polygon(B, A, 4) → poly3, j, k, l, m, H, I
  8. Angle(A, C, B) → α
  9. Midpoint(C, F) → J
  10. Midpoint(C, E) → K
  11. Line(J, j) → n
  12. Line(K, j) → p
  13. OrthogonalLine(J, j) → q
  14. OrthogonalLine(K, j) → r
  15. Intersect(n, i) → L
  16. Intersect(q, f) → M
  17. Intersect(n, g) → N
  18. Intersect(q, h) → O
  19. Intersect(r, d) → P
  20. Intersect(p, e) → Q
  21. Intersect(p, b) → R
  22. Intersect(r, a) → S
  23. Polygon(O, G, L, J) → poly4, o, g_1, l_1, j_1
  24. Polygon(L, C, M, J) → poly5, l_2, c_1, m_1, j_2
  25. Polygon(O, J, N, F) → poly6, o_1, j_3, n_1, f_1
  26. Polygon(J, M, B, N) → poly7, j_4, m_2, b_1, n_2
  27. Polygon(P, K, Q, E) → poly8, p_1, k_1, q_1, e_1
  28. Polygon(Q, A, S, K) → poly9, q_2, a_1, s, k_2
  29. Polygon(K, S, C, R) → poly10, k_3, s_1, c_2, r_1
  30. Polygon(K, P, D, R) → poly11, k_4, p_2, d_1, r_2
  31. Midpoint(H, B) → T
  32. If(t ≤ 1, t, 1) → t1
  33. If(t ≤ 1, 0, If[t ≤ 2, t - 1, 1]) → t2
  34. If(t ≤ 2, 0, If[t ≤ 3, t - 2, 1]) → t3
  35. If(t ≤ 3, 0, If[t ≤ 4, t - 3, 1]) → t4
  36. If(t ≤ 4, 0, If[t ≤ 5, t - 4, 1]) → t5
  37. Translate(Element[list1, 5], Vector[(t5 * Vector[K, T])]) → list1e
  38. Translate(Element[list1, 1], Vector[(t1 * Vector[J, I])]) → list1a
  39. Translate(Element[list1, 2], Vector[(t2 * Vector[J, B])]) → list1b
  40. Translate(Element[list1, 4], Vector[(t3 * Vector[J, A])]) → list1d
  41. Translate(Element[list1, 3], Vector[(t4 * Vector[J, H])]) → list1c
  42. Translate(Element[list2, 1], Vector[(t1 * Vector[K, I])]) → list2a
  43. Translate(Element[list2, 2], Vector[(t2 * Vector[K, B])]) → list2b
  44. Translate(Element[list2, 3], Vector[(t3 * Vector[K, A])]) → list2c
  45. Translate(Element[list2, 4], Vector[(t4 * Vector[K, H])]) → list2d
  46. Translate(Element[list2, 5], Vector[(t5 * Vector[J, T])]) → list2e

AI Drawing Prompt Reference

Draw a figure composed of two semicircles that can be dissected and rearranged to form a square. Start by creating points A and B on the x-axis, then draw two semicircles with A and B as their diameters. Next, create four vertices on each semicircle to form three quadrilaterals. Find key angles and midpoints, then draw several lines and perpendiculars using these points. Finally, rearrange the quadrilaterals through translation to form a square.
The prompt above was compiled from the original drawing commands and is for reference only.