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Right Triangle Wonder: Geometric Construction and Proof

Right Triangle Wonder: Geometric Construction and Proof
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2020-09-02 09:32
TriangleCirclePolygonAngleGeometric Construction
This GeoGebra example dynamically demonstrates the geometric properties and proof ideas of right triangles. Core concepts include right-triangle geometry, angle relationships, polygon construction, circles and circle sectors, similarity transformations such as rotation and dilation, and intersection operations among points, lines, and circles. Construction highlights: it uses commands Point, Segment, Polygon, Angle, Circle, CircleSector, Intersect, Rotate, Dilate, Midpoint, Line, Ray, and Min, along with point objects A through H, angle measures α, β, γ, δ, and geometric objects poly1, c, c_1. Suitable for middle-school and high-school plane geometry teaching, classroom demonstration, independent student inquiry, and geometry proof assistance.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(Ray[(0, 0), (1, 0)]) → B
  3. Point(Ray[(0, 0), (0, 1)]) → C
  4. Polygon(A, B, C) → t1, c, a, b
  5. Segment(C, A) → f
  6. Segment(A, B) → g
  7. Segment(B, C) → h
  8. Polygon(C, B, 4) → poly1, i, j, k, l, D, E
  9. Rotate(B, 45°, A) → B'
  10. Angle(B, A, B') → α
  11. Ray(A, B') → m
  12. Line(E, D) → n
  13. Intersect(n, m) → F
  14. Intersect(h, m) → G
  15. Polygon(C, G, F, E) → q1, c_1, g_1, f_1, e
  16. Midpoint(G, F) → H
  17. Segment(B, D) → p
  18. Segment(D, E) → q
  19. Segment(E, C) → r
  20. Angle(C, E, D) → β
  21. Angle(E, D, B) → γ
  22. Angle(D, B, C) → δ
  23. Dilate(A, If[0 < d ≤ 1, 1 - d, 0], G) → A'
  24. Segment(A, A') → s
  25. Dilate(G, If[1 < d ≤ 2, 2 - d, 0], F) → G'
  26. Segment(G, G') → t
  27. Min(g, f) → o
  28. Circle(A, (0.5 * o)) → d_1
  29. Intersect(d_1, g, 1) → I
  30. Rotate(I, If[3 < d ≤ 4, ((d - 3) * π / 4), π / 4], A) → I'
  31. CircleSector(A, I, I') → e_1
  32. Rotate(e_1, If[5 < d ≤ 6, ((d - 5) * π / 4), π / 4], A) → h_1
  33. Rotate(C, If[7 < d ≤ 8, ((-(d - 7)) * π), π], H) → C'
  34. Rotate(G, If[7 < d ≤ 8, ((-(d - 7)) * π), π], H) → G'_1
  35. Rotate(F, If[7 < d ≤ 8, ((-(d - 7)) * π), π], H) → F'
  36. Rotate(E, If[7 < d ≤ 8, ((-(d - 7)) * π), π], H) → E'
  37. Polygon(C', G'_1, F', E') → q1', c', g'_1, f', e'
  38. Angle(B, A, C) → ε
  39. Ray((0, 0), (1, 0)) → i_1
  40. Ray((0, 0), (0, 1)) → j_1