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Dynamic Proof of the Law of Sines for Acute Angle

Dynamic Proof of the Law of Sines for Acute Angle
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2012-10-26 18:39
AngleTrigonometryGeometric Construction
In collections Grade 12 Calculus & Vectors MCV4U1
This example dynamically demonstrates the law of sines for acute angles. It involves the concept of trigonometry, specifically sine. Using GeoGebra commands such as Circle, Point, Polygon, and operations like rotation and intersection, a triangle is constructed to visually prove the law of sines. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Circle(O, P) → c
  2. Point(c) → A
  3. Point(c) → B
  4. Point(c) → C
  5. Polygon(A, B, C) → mnohoúhelník1, e, c_1, d
  6. Angle(B, A, C) → α_1
  7. Angle(A, C, O) → β
  8. If(a < 1, a, 1) → alfa
  9. Line(C, A) → b
  10. Rotate(b, alfa * β, C) → b'
  11. Intersect(c, b') → D, E
  12. Angle(C, E, B) → γ
  13. Angle(B, E, C) → α_2
  14. Polygon(B, C, E) → mnohoúhelník2, e_1, b_1, c_2
  15. Angle(C, B, E) → α
  16. If(a < 1, 0, If[a < 2, a - 1, 1]) → beta
  17. Rotate(B, (beta * 90)°, F) → B'
  18. Rotate(C, (beta * 90)°, F) → C'
  19. Rotate(E, (beta * 90)°, F) → E'
  20. Polygon(B', C', E') → mnohoúhelník2', e', b'_1, c'
  21. Angle(C', B', E') → δ
  22. Angle(B', E', C') → ε
  23. Angle(C, B, A) → ζ
  24. Angle(A, C, B) → η
  25. Line(A, B) → i
  26. Angle(B, A, O) → θ
  27. Rotate(i, alfa * θ, A) → i'
  28. Intersect(c, i') → G, H
  29. Polygon(C, A, H) → mnohoúhelník3, h_1, c_3, a_1
  30. Angle(C, H, A) → ι
  31. Angle(A, C, H) → κ
  32. Rotate(C, -((beta * 35)°), I) → C'_1
  33. Rotate(A, -((beta * 35)°), I) → A'
  34. Rotate(H, -((beta * 35)°), I) → H'
  35. Polygon(C'_1, A', H') → mnohoúhelník3', h', c'_1, a'
  36. Angle(C'_1, H', A') → λ
  37. Angle(A', C'_1, H') → μ
  38. Line(B, C) → j
  39. Angle(C, B, O) → ν
  40. Rotate(j, alfa * ν, B) → j'
  41. Intersect(c, j') → J, K
  42. Polygon(A, B, K) → mnohoúhelník4, k, a_2, b_2
  43. Angle(B, A, K) → ξ
  44. Angle(A, K, B) → ο
  45. Rotate(A, -((beta * 147)°), L) → A'_1
  46. Rotate(B, -((beta * 147)°), L) → B'_1
  47. Rotate(K, -((beta * 147)°), L) → K'
  48. Polygon(A'_1, B'_1, K') → mnohoúhelník4', k', a'_1, b'_2
  49. Angle(B'_1, A'_1, K') → ρ
  50. Angle(A'_1, K', B'_1) → σ

AI Drawing Prompt Reference

Draw a dynamic illustration of the Law of Sines for acute angles. Start by drawing a circle and marking three points on it to form a triangle, labeling the angles. Then, use rotation and intersection to visually demonstrate the Law of Sines, showing the relationship between angles and side lengths. Include elements like triangles, rotated lines, and intersection points to highlight the geometric proof.
The prompt above was compiled from the original drawing commands and is for reference only.