MathPlotAI Logo MathPlotAI

Euler's Theorem: Multiple Geometric Demonstrations (Triangle, Circles, and Const

Euler's Theorem: Multiple Geometric Demonstrations (Triangle, Circles, and Const
Loading the math board and drawing, please wait…
2019-08-01 16:48
TriangleCirclePolygonAngleGeometric Construction
This GeoGebra example, titled 'Euler (démonstrations)', provides a comprehensive dynamic visualization of Euler's theorem in triangle geometry. It demonstrates the collinearity of the centroid, circumcenter, orthocenter, and the nine-point center (Euler line), along with the construction of related circles such as the circumcircle, incircle, and excircles. The construction involves core mathematical concepts including triangle centers, perpendicular bisectors, angle bisectors, medians, altitudes, and circle properties. Using GeoGebra commands such as Line, OrthogonalLine, CircumcircleArc, Segment, Intersect, Circle, Angle, Polygon, Midpoint, Ray, LineBisector, and AngularBisector, the example builds multiple objects (points A, B, C, H, O, etc.) and auxiliary circles/polygons to illustrate the relationships. It is suitable for high school and university students or teachers exploring classical geometry theorems, and serves as an advanced exercise in dynamic geometric construction and theorem verification.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. OrthogonalLine(C, a) → d
  5. C_Euler(A, B, C) → D_1, e, B', A', C'
  6. Point(e) → F
  7. Point(e) → G_1
  8. CircumcircleArc(G_1, A', F) → g
  9. Segment(C', A') → h
  10. Segment(B', C) → i
  11. Segment(B', A) → j
  12. Intersect(a, d) → H_c
  13. Segment(H_c, B') → k
  14. Circle(B', H_c) → p
  15. Angle(C, H_c, A) → α
  16. Polygon(B', A', C', H_c) → poly1, b', a', c', h_c
  17. OrthogonalLine(B, c) → n
  18. Intersect(d, n) → H
  19. Midpoint(H, C) → H_3
  20. Polygon(A, H, C) → poly2, c_1, a_1, h_1
  21. Ray(B', H_3) → q
  22. Intersect(b, q) → I_1
  23. Ray(H_3, A') → r
  24. Ray(H_3, C') → s
  25. Segment(C', B') → t
  26. Angle(C', B', H_3) → β
  27. Point(e) → J
  28. Point(e) → K
  29. CircumcircleArc(J, H_c, K) → d_1
  30. Ray(A, H) → b_1
  31. Intersect(b, b_1) → L
  32. Angle(C, L, A) → γ
  33. Cercle_3pts(A, C, B) → O, e_1
  34. Line(A, O) → f_1
  35. Intersect(e_1, f_1, 2) → D
  36. Segment(C, D) → g_1
  37. Segment(D, B) → i_1
  38. Angle(A, C, D) → δ
  39. Angle(D, B, A) → ε
  40. OrthogonalLine(B, c) → j_1
  41. Polygon(B, H, C, D) → poly3, b_2, h_2, c_2, d_2
  42. Segment(H, D) → k_1
  43. Segment(A, A') → l_1
  44. TriPart(A, A') → G, M
  45. Polygon(A, H, D) → poly4, d_3, a_2, h_3
  46. Segment(H, O) → m_1
  47. Polygon(C, H, O, C') → poly5, c_3, h_4, o, c'_1
  48. OrthogonalLine(A, b) → n_1
  49. OrthogonalLine(B, h_1) → p_1
  50. Intersect(c, p_1) → H_b
  51. Intersect(b, n_1) → H_a
  52. LineBisector(H_c, C') → q_1
  53. Segment(H, O) → r_1
  54. Segment(O, C') → s_1
  55. Segment(O, A') → t_1
  56. Segment(O, B') → e_2
  57. Segment(H, H_c) → f_2
  58. Segment(H, H_b) → g_2
  59. Segment(H, H_a) → i_2
  60. LineBisector(H_a, A') → j_2
  61. LineBisector(H_b, B') → k_2
  62. Intersect(d_1, b_1) → H_1, Q
  63. Midpoint(H, B) → H_2
  64. C_Inscrit(A, C, B) → I, p_2
  65. AngularBisector(h_1, b) → l_2, m_2
  66. AngularBisector(b, a) → n_2, q_2
  67. Intersect(l_2, q_2) → E
  68. Intersect(l_2, n_2) → N
  69. Intersect(m_2, n_2) → P
  70. OrthogonalLine(P, a) → r_2
  71. OrthogonalLine(N, b) → s_2
  72. OrthogonalLine(E, h_1) → t_2
  73. Intersect(a, r_2) → R
  74. Intersect(s_2, b) → S
  75. Intersect(t_2, c) → T
  76. Circle(N, S) → e_3
  77. Circle(P, R) → f_3
  78. Circle(E, T) → g_3
  79. Intersect(e, g_3) → F_2, , ,
  80. Intersect(e, e_3) → F_1, , ,
  81. Intersect(e, f_3) → F_3, , ,
  82. Intersect(e, p_2) → F_e, , ,
  83. Intersect(g_3, b, 2) → U
  84. Intersect(g_3, a, 2) → V
  85. Intersect(f_3, b, 1) → A_1
  86. Intersect(e_3, a, 1) → B_1
  87. Intersect(e_3, c, 2) → W
  88. Intersect(f_3, c, 1) → Z
  89. Segment(A, B) → a_3
  90. Segment(B, C) → b_3
  91. Segment(C, A) → i_3

AI Drawing Prompt Reference

Draw a triangle with its three side lines extended. From one vertex, drop a perpendicular to the opposite side, marking the foot. Construct the circumcircle of the triangle and label its center. Draw segments from the orthocenter to a point on the circumcircle)Skip that; instead, connect the orthocenter to the triangle's vertices and the circumcenter. Construct a midline of the triangle and mark its midpoint. Inside the triangle, form a polygon using the orthocenter, circumcenter, and a vertex. Then draw a ray from a vertex intersecting the opposite side. Finally, label several key angles and add auxiliary circles tangent to the triangle's sides.
The prompt above was compiled from the original drawing commands and is for reference only.