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Law of Cosines Demonstration by John Golden

Law of Cosines Demonstration by John Golden
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2014-11-15 02:39
TriangleCircleTrigonometryCoordinate GeometryGeometric Construction
This example demonstrates the Law of Cosines in an arbitrary triangle, showing the relationship between side lengths and angles dynamically. It involves core mathematical concepts such as the Law of Cosines and its connection to the Pythagorean Theorem. The construction uses GeoGebra commands and objects like Polygon, Line, OrthogonalLine, Intersect, Segment, Angle, Ray, Rotate, Circle, Textfield, and Midpoint. It is suitable for high school mathematics education, helping students visually understand the proof and application of the Law of Cosines.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Line(B, A) → d
  3. Line(C, A) → e
  4. OrthogonalLine(C, d) → f
  5. OrthogonalLine(B, e) → g
  6. OrthogonalLine(A, a) → h
  7. Intersect(h, a) → D
  8. Intersect(d, f) → E
  9. Intersect(e, g) → F
  10. Polygon(C, B, 4) → poly2, i, j, k, l, G, H
  11. Polygon(A, C, 4) → poly3, m, n, p, q, I, J
  12. Polygon(B, A, 4) → poly4, r, s, t, a_1, K, L
  13. Line(L, K) → b_1
  14. Line(I, J) → c_1
  15. Intersect(f, b_1) → M
  16. Intersect(g, c_1) → N
  17. Polygon(A, E, M, K) → poly5, a_2, e_1, m_1, k_1
  18. Polygon(A, F, N, J) → poly6, a_3, f_1, n_1, j_1
  19. Intersect(h, k) → O
  20. Polygon(D, C, H, O) → poly7, d_1, c_2, h_1, o
  21. Polygon(D, O, G, B) → poly8, d_2, o_1, g_1, b_2
  22. Polygon(B, E, M, L) → poly9, b_3, e_2, m_2, l_1
  23. Polygon(C, I, N, F) → poly10, c_3, i_1, n_2, f_2
  24. Segment(A, D) → p_1
  25. Segment(C, E) → q_1
  26. Segment(B, F) → r_1
  27. Angle(B, A, C) → α
  28. Segment(A, E) → s_1
  29. Segment(A, F) → t_1
  30. Ray(B, C) → g_2
  31. Ray(C, B) → h_2
  32. Rotate(C, (-((180° - setalpha) / 2)), C) → C'
  33. Rotate(B, (-((180° - (2 * setalpha)) / 2)), C) → B'
  34. Ray(C', B') → i_2
  35. Rotate(B, (180° - (2 * setalpha)) / 2, B) → B'_1
  36. Rotate(C, (180° - (2 * setalpha)) / 2, B) → C'_1
  37. Ray(B'_1, C'_1) → j_2
  38. Intersect(i_2, j_2) → P
  39. Circle(P, C) → k_2
  40. Ray(D, A) → l_2
  41. Intersect(k_2, l_2) → Q,
  42. Textfield(setalpha) → inputBox1
  43. Midpoint(B, C) → R
  44. Circle(R, C) → p_2
  45. Intersect(p_2, l_2) → S,

AI Drawing Prompt Reference

Draw a geometric figure demonstrating the Law of Cosines. The figure should include an arbitrary triangle, and elements that show the relationship between side lengths and angles. Key objects include triangles, line segments, angles, rays, circles, and a text box. The construction order should start with drawing the triangle, then adding related line segments and angles, and finally using rotations and circle constructions to show the theorem's relationship.
The prompt above was compiled from the original drawing commands and is for reference only.