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Proof of the Law of Sines by Irina Boyadzhiev

Proof of the Law of Sines by Irina Boyadzhiev
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2016-07-18 15:27
TriangleTrigonometryGeometric Construction
This example demonstrates the proof of the Law of Sines, which relates the lengths of the sides of a triangle to the sines of its angles. Using objects like Circle, Point, Polygon, Angle, Segment, Mirror, CircleArc, and If, the geometric proof is illustrated. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Circle(O, Z) → c1
  2. Point(c1) → A
  3. Point(c1) → B
  4. Point(c1) → C
  5. Polygon(A, B, C) → poly1, c, a, b
  6. Angle(poly1) → α, β, γ
  7. Segment(B, O) → h
  8. Segment(O, C) → j
  9. Segment(O, A) → i
  10. Mirror(B, O) → F
  11. Mirror(C, O) → H
  12. Mirror(A, O) → J
  13. CircleArc(O, C, F) → e
  14. CircleArc(O, A, H) → p
  15. CircleArc(O, B, J) → r
  16. Point(e) → M1
  17. Point(e, 0) → M2
  18. If(sh_j ≟ false ∧ sh_i ≟ false, M1, C) → M
  19. Polygon(A, B, M) → poly2, m, a_1, b_1
  20. Angle(A, M, B) → γ_1
  21. Angle(B, A, M) → δ
  22. Point(p) → N1
  23. Point(p, 0) → N2
  24. If(sh_h ≟ false ∧ sh_i ≟ false, N1, N2) → N
  25. Polygon(N, B, C) → poly3, c_1, n, b_2
  26. Angle(B, N, C) → α_1
  27. Angle(C, B, N) → ε
  28. Point(r) → P1
  29. Point(r, 0) → P2
  30. If(sh_h ≟ false ∧ sh_j ≟ false, P1, P2) → P
  31. Polygon(A, P, C) → poly4, c_2, a_2, p_1
  32. Angle(C, P, A) → β_1
  33. Angle(A, C, P) → η

AI Drawing Prompt Reference

Draw a circle with center O. Place three points A, B, C on the circle and connect them to form a triangle. Draw radii from O to each vertex. Reflect each vertex across O to get three opposite points. Construct three circular arcs from O through each vertex and its reflection. On each arc, choose a point and connect it to the triangle vertices to form three additional triangles. Label the original triangle's angles and the related angles in the new triangles to illustrate the Law of Sines proof.
The prompt above was compiled from the original drawing commands and is for reference only.