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Relationship Between Two Equilateral Triangles on Two Sides of Any Triangle

Relationship Between Two Equilateral Triangles on Two Sides of Any Triangle
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2017-08-18 19:53
Triangle
This example illustrates a special relationship that occurs when two equilateral triangles are constructed on two sides of any triangle. Key mathematical concepts include equilateral triangles and congruence. Construction highlights the use of GeoGebra commands such as Segment, Rotate, Polygon, etc. Suitable for middle school and above students to learn about geometric transformations and congruence of triangles.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Rotate(B, (a * π / 180), A) → B'
  3. Angle(B, A, B') → α
  4. Ray(A, B') → g
  5. Point(g) → C
  6. Segment(B, C) → h
  7. Segment(A, C) → i
  8. Polygon(A, B, C) → poly1, c, a_1, b
  9. Polygon(C, B, 3) → poly2, j, k, l, D
  10. Polygon(A, C, 3) → poly3, m, n, p, E
  11. Midpoint(C, E) → F
  12. Midpoint(C, D) → G
  13. Midpoint(A, B) → H
  14. Rotate(A, If[0 < d ≤ 1, (d * π / 3), π / 3], A) → A'
  15. Rotate(C, If[0 < d ≤ 1, (d * π / 3), π / 3], A) → C'
  16. Segment(A', C') → i'
  17. Rotate(B, If[0 < d ≤ 1, (d * π / 3), π / 3], C) → B'_1
  18. Rotate(C, If[0 < d ≤ 1, (d * π / 3), π / 3], C) → C'_1
  19. Segment(B'_1, C'_1) → h'
  20. Rotate(B'_1, If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], D) → I
  21. Rotate(C'_1, If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], D) → J
  22. Segment(I, J) → h''
  23. Rotate(A', If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], E) → A''
  24. Rotate(C', If[1 < d ≤ 2, ((d - 1) * π / 3), π / 3], E) → C''
  25. Segment(A'', C'') → i''
  26. Dilate(A, If[2 < d ≤ 3, 3 - d, 0], H) → A'_1
  27. Segment(A, A'_1) → q
  28. Rotate(A, If[3 < d ≤ 4, ((-(d - 3)) * π), (-π)], H) → A'_2
  29. Rotate(A'_1, If[3 < d ≤ 4, ((-(d - 3)) * π), (-π)], H) → K
  30. Segment(A'_2, K) → q'
  31. Dilate(E, If[4 < d ≤ 5, 5 - d, 0], F) → E'
  32. Dilate(D, If[4 < d ≤ 5, 5 - d, 0], G) → D'
  33. Segment(E, E') → r
  34. Segment(D, D') → s
  35. Rotate(E, If[5 < d ≤ 6, ((d - 5) * π), π], F) → E'_1
  36. Rotate(E', If[5 < d ≤ 6, ((d - 5) * π), π], F) → E''
  37. Segment(E'_1, E'') → r'
  38. Rotate(D, If[5 < d ≤ 6, ((-(d - 5)) * π), (-π)], G) → D'_1
  39. Rotate(D', If[5 < d ≤ 6, ((-(d - 5)) * π), (-π)], G) → D''
  40. Segment(D'_1, D'') → s'
  41. Polygon(G, H, F) → poly4, f_1, g_1, h_1
  42. Dilate(G, If[6 < d ≤ 7, 7 - d, 0], H) → G'
  43. Segment(G, G') → t
  44. Rotate(G, If[7 < d ≤ 8, ((d - 7) * π / 3), π / 3], H) → G'_1
  45. Rotate(G', If[7 < d ≤ 8, ((d - 7) * π / 3), π / 3], H) → G''
  46. Segment(G'_1, G'') → t'
  47. Rotate(G'_1, If[8 < d ≤ 9, ((d - 8) * π / 3), π / 3], F) → L
  48. Rotate(G'', If[8 < d ≤ 9, ((d - 8) * π / 3), π / 3], F) → G'''
  49. Segment(L, G''') → t''
  50. Rotate(G'_1, If[8 < d ≤ 9, ((d - 8) * π / 3), π / 3], F) → M
  51. Rotate(G'', If[8 < d ≤ 9, ((d - 8) * π / 3), π / 3], F) → G'''_1
  52. Segment(M, G'''_1) → t''_1