MathPlotAI Logo MathPlotAI

Cosine Law by Earl Samuelson

Cosine Law by Earl Samuelson
Loading the math board and drawing, please wait…
2015-12-28 02:05
TriangleTrigonometryGeometric Construction
This example illustrates the Cosine Law, which states that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice their product with the cosine of the included angle. Key concepts include trigonometry and geometric construction. It uses commands like Circle, Intersect, Polygon, Angle, OrthogonalLine, Segment, Line, PointIn, Midpoint in GeoGebra.

GeoGebra Commands & Parameters

  1. Circle(A_1, 5) → c_4
  2. Intersect(c_4, b_1, 1) → A
  3. Intersect(c_4, b_1, 2) → B
  4. Intersect(c_4, d, 1) → C
  5. Polygon(A, C, B) → poly1, b_2, a_1, c_1
  6. Angle(A, C, B) → α
  7. OrthogonalLine(B, b_2) → e
  8. OrthogonalLine(C, c_1) → f
  9. OrthogonalLine(A, a_1) → g_3
  10. Intersect(e, b_2) → D
  11. Intersect(g_3, a_1) → E
  12. Intersect(f, c_1) → F
  13. Circle(F, c_1) → h_3
  14. Intersect(h_3, f, 2) → G
  15. Segment(F, G) → i
  16. Circle(D, b_2) → k
  17. Intersect(k, e, 2) → H
  18. Segment(D, H) → j
  19. Circle(E, a_1) → p
  20. Intersect(p, g_3, 2) → I
  21. Segment(E, I) → l
  22. Line(H, b_2) → m
  23. Line(G, c_1) → n
  24. Line(I, a_1) → q
  25. Line(C, j) → r
  26. Line(A, j) → s
  27. Line(A, i) → t
  28. Line(B, i) → d_1
  29. Line(B, l) → e_1
  30. Line(C, l) → f_1
  31. Intersect(m, r) → J
  32. Intersect(m, s) → K
  33. Intersect(q, f_1) → L
  34. Intersect(q, e_1) → M
  35. Intersect(n, t) → N
  36. Intersect(n, d_1) → O
  37. Polygon(D, H, J, C) → poly2, d_2, h_1, j_1, c_2
  38. Polygon(C, L, I, E) → poly3, c_3, l_1, i_1, e_2
  39. PointIn(poly2) → Pointpoly2
  40. PointIn(poly3) → Pointpoly3_1
  41. Polygon(A, K, H, D) → poly4, a_2, k_1, h_2, d_3
  42. Polygon(A, F, G, N) → poly5, a_3, f_2, g_1, n_1
  43. PointIn(poly5) → Pointpoly5
  44. PointIn(poly4) → Pointpoly4
  45. Polygon(F, B, O, G) → poly6, f_3, b_3, o, g_2
  46. Polygon(B, M, I, E) → poly7, a_5, m_1, i_2, e_3
  47. PointIn(poly7) → Pointpoly7
  48. PointIn(poly6) → Pointpoly6
  49. PointIn(poly3) → Pointpoly3
  50. Segment(B, D) → p_1
  51. Segment(A, E) → q_1
  52. Segment(F, C) → r_1
  53. Midpoint(h_1) → P
  54. Midpoint(k_1) → Q
  55. Midpoint(g_1) → R
  56. Midpoint(o) → S
  57. Midpoint(l_1) → T
  58. Midpoint(m_1) → U
  59. Segment(P, V) → s_1
  60. Segment(Q, W) → t_1
  61. Segment(R, Z) → j_2
  62. Segment(S, B_1) → k_2
  63. Segment(U, C_1) → l_2
  64. Segment(T, D_1) → m_2
  65. Segment(C, J) → b
  66. Segment(B, M) → a
  67. Segment(A, N) → c

AI Drawing Prompt Reference

Draw a triangle ABC with A, B, and C as the intersection points of circles with radius 5 and lines b1, d. Construct the sine lines for angles A, B, and C, then use perpendicular lines to create auxiliary segments, forming shapes such as quadrilateral DHJC, triangle FBI, and triangle EIC, each containing specific points and segments.
The prompt above was compiled from the original drawing commands and is for reference only.