MathPlotAI Logo MathPlotAI

Orthocenter and Circumcircle Dynamic Construction

Orthocenter and Circumcircle Dynamic Construction
Loading the math board and drawing, please wait…
2016-10-21 06:40
TriangleCircleGeometric ConstructionLocus
This example demonstrates the geometric construction of the orthocenter of a triangle and its relationship with the circumcircle. The core mathematical concepts include the orthocenter (intersection of the three altitudes), the circumcircle (circle passing through the three vertices), and the perpendicularity of altitudes. Construction details: using GeoGebra commands such as Segment, Point, Circle, Intersect, OrthogonalLine, and Line, the triangle ABC is built, then perpendicular lines to each side are drawn to find the orthocenter D, and the circumcircle c is constructed. Additional objects (feet of altitudes, extended lines, etc.) are generated to create an interactive dynamic figure. Suitable for middle and high school geometry classes, this helps students visualize the orthocenter's position for acute, right, and obtuse triangles, and explore properties like the Euler line or the nine-point circle.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Segment(B, C) → g
  3. Segment(A, C) → h
  4. Point(h) → D
  5. Circle(B, D) → c
  6. Intersect(c, h, 2) → E
  7. Point(h) → F
  8. Circle(E, F) → d
  9. Segment(A_1, B_1) → f_1
  10. Segment(B_1, C_1) → g_1
  11. Segment(A_1, C_1) → h_1
  12. Point(h_1) → D_1
  13. Circle(B_1, D_1) → c_1
  14. Intersect(c_1, h_1, 2) → E_1
  15. Segment(A_2, B_2) → f_2
  16. Segment(B_2, C_2) → g_2
  17. Segment(A_2, C_2) → h_2
  18. Point(h_2) → D_2
  19. Circle(B_2, D_2) → c_2
  20. Intersect(c_2, h_2, 2) → E_2
  21. Point(h_2) → F_1
  22. Circle(E_2, F_1) → d_1
  23. Intersect(d, d_1, 1) → G
  24. Intersect(d, d_1, 2) → H
  25. Segment(B, H) → i
  26. Segment(H, B) → j
  27. OrthogonalLine(F, i) → k
  28. Intersect(k, h) → I
  29. Intersect(k, h) → J
  30. Point(i) → K
  31. OrthogonalLine(K, h) → l
  32. Point(f) → L
  33. Line(B, L) → m
  34. Circle(F_1, M) → e
  35. Intersect(e, m, 2) → N
  36. Intersect(e, m, 1) → O
  37. Segment(A_3, B_3) → f_3
  38. Segment(B_3, C_3) → g_3
  39. Segment(A_3, C_3) → h_3
  40. Point(h_3) → D_3
  41. Circle(B_3, D_3) → c_3
  42. Point(h_3) → F_2
  43. Segment(A_4, B_4) → f_4
  44. Segment(B_4, C_4) → g_4
  45. Segment(A_4, C_4) → h_4
  46. Point(h_4) → D_4
  47. Circle(B_4, D_4) → c_4
  48. Point(h_4) → F_3
  49. Point(c_4) → P
  50. Point(c_4) → Q
  51. Segment(F_1, Q) → n
  52. Segment(Q, P) → p
  53. Segment(F_1, P) → q
  54. Intersect(d_1, q, 2) → R
  55. Line(R, P) → r
  56. Point(l) → S
  57. Point(l) → T
  58. Line(S, T) → s
  59. Point(g) → U
  60. Point(g) → V
  61. Line(U, V) → t
  62. Segment(A_5, C_5) → h_5
  63. Point(h_5) → D_5
  64. Point(h_5) → F_4
  65. Circle(F_4, M_1) → e_1
  66. Segment(A_6, B_5) → f_5
  67. Segment(B_5, C_6) → g_5
  68. Segment(A_6, C_6) → h_6
  69. Point(h_6) → D_6
  70. Circle(B_5, D_6) → c_5
  71. Point(h_6) → F_5
  72. Segment(A_7, B_7) → f_6
  73. Segment(B_7, C_7) → g_6
  74. Segment(A_7, C_7) → h_7
  75. Point(h_7) → D_7
  76. Circle(B_7, D_7) → c_6
  77. Point(h_7) → F_6
  78. Point(c_5) → W
  79. Point(c_5) → Z
  80. Point(c_5) → G_1
  81. Line(W, G_1) → a
  82. Intersect(l, r) → H_1
  83. Point(a) → I_1
  84. Segment(I_1, H_1) → b
  85. Segment(F_1, I_1) → i_1
  86. Segment(H_1, F_1) → j_1
  87. Segment(I_1, H_1) → k_1
  88. Segment(K, H_1) → l_1
  89. Segment(B, F_4) → m_1