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Geometry Construction Example: Triangle and Centers

Geometry Construction Example: Triangle and Centers
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2019-04-08 20:54
TriangleCircleAngleGeometric Construction
This example demonstrates how to construct a triangle and find its orthocenter, circumcenter, and centroid using GeoGebra commands. It involves core math concepts like triangles, angles, similar triangles, equilateral triangles, LCM, and GCD. Key GeoGebra commands and objects used include Polygon, OrthogonalLine, Intersect, Midpoint, Segment, Distance, LaTeX, Line, Angle, Rotate, Centroid, InteriorAngles, and Text. Suitable for high school students learning geometric constructions and properties of triangle centers.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → t1, c, a, b
  2. OrthogonalLine(A, a) → f
  3. OrthogonalLine(C, c) → g
  4. OrthogonalLine(B, b) → h
  5. Intersect(f, h) → Orthocenter
  6. Midpoint(c) → D
  7. Midpoint(b) → E
  8. Midpoint(a) → F
  9. OrthogonalLine(F, a) → i
  10. OrthogonalLine(E, b) → j
  11. OrthogonalLine(D, c) → k
  12. Intersect(j, i) → Circumcenter
  13. Segment(A, F) → l
  14. Segment(B, E) → m
  15. Segment(C, D) → n
  16. Intersect(l, n) → Centroid
  17. Segment(Orthocenter, Circumcenter) → p
  18. Distance(Orthocenter, Centroid) → distanceOrthocenterCentroid
  19. Distance(Centroid, Circumcenter) → distanceCentroidCircumcenter
  20. LaTeX(d, true, true) → text1
  21. Polygon(G, H, I) → t2, i_1, g_1, h_1
  22. Midpoint(h_1) → J
  23. Midpoint(i_1) → K
  24. Midpoint(g_1) → L
  25. Polygon(J, K, L) → t3, l_1, j_1, k_1
  26. Polygon(M, N, O) → t4, o, m_1, n_1
  27. Distance(M, P) → distanceMP
  28. Distance(P, N) → distancePN
  29. Distance(P, O) → distancePO
  30. LaTeX(e, true, true) → text2
  31. LaTeX(e, true, true) → text3
  32. Midpoint(n_1) → Q
  33. Midpoint(o) → R
  34. Midpoint(m_1) → S
  35. Polygon(T, U, V) → t5, v, t, u
  36. Polygon(T, U, 3) → poly1, q, r, s, W
  37. Polygon(V, T, 3) → poly2, f_1, p_1, q_1, Z
  38. Segment(Z, U) → r_1
  39. Segment(W, V) → s_1
  40. LaTeX(text5, true, true) → text6
  41. Polygon(A_1, B_1, C_1) → t6, c_1, a_1, b_1
  42. OrthogonalLine(B_1, b_1) → t_1
  43. OrthogonalLine(C_1, c_1) → d_1
  44. OrthogonalLine(A_1, a_1) → e_1
  45. Intersect(a_1, e_1) → D_1
  46. Intersect(t_1, b_1) → E_1
  47. Intersect(c_1, d_1) → F_1
  48. Polygon(F_1, D_1, E_1) → t7, f_2, g_2, h_2
  49. Polygon(G_1, H_1, I_1) → t8, i_2, j_2, k_2
  50. OrthogonalLine(J_1, k_2) → l_2
  51. OrthogonalLine(J_1, j_2) → m_2
  52. OrthogonalLine(J_1, i_2) → n_2
  53. Intersect(l_2, k_2) → K_1
  54. Intersect(n_2, i_2) → L_1
  55. Intersect(m_2, j_2) → M_1
  56. Polygon(L_1, M_1, K_1) → t9, p_2, q_2, r_2
  57. Polygon(N_1, O_1, P_1) → t10, s_2, t_2, o_1
  58. Line(N_1, Q_1) → a_2
  59. Line(P_1, Q_1) → b_2
  60. Line(O_1, Q_1) → c_2
  61. Intersect(b_2, s_2) → R_1
  62. Intersect(a_2, t_2) → S_1
  63. Intersect(c_2, o_1) → T_1
  64. Polygon(R_1, T_1, S_1) → t11, d_2, e_2, f_3
  65. Polygon(U_1, V_1, W_1) → t12, w_1, u_1, v_1
  66. Angle(U_1, V_1, W_1) → α
  67. Rotate(U_1, (1 / 3 * α), V_1) → A_2
  68. Rotate(W_1, (-((1 / 3 * α))), V_1) → Z_1
  69. Angle(W_1, U_1, V_1) → β
  70. Rotate(W_1, (1 / 3 * β), U_1) → B_{2}
  71. Rotate(V_1, (-1 / 3 * β), U_1) → C_2
  72. Angle(V_1, W_1, U_1) → γ
  73. Polygon(D_2, E_2, F_2) → t13, g_3, h_3, i_3
  74. Polygon(D_2, E_2, 3) → poly3, j_3, k_3, l_3, G_2
  75. Polygon(E_2, F_2, 3) → poly4, m_3, n_3, p_3, H_2
  76. Polygon(F_2, D_2, 3) → poly5, q_3, r_3, s_3, I_2
  77. Centroid(F_2, D_2, I_2) → J_2
  78. Centroid(H_2, E_2, F_2) → K_2
  79. Centroid(E_2, D_2, G_2) → L_2
  80. Segment(L_2, F_2) → t_3
  81. Segment(E_2, J_2) → b_3
  82. Segment(K_2, D_2) → a_3
  83. Intersect(b_3, a_3) → M_2
  84. InteriorAngles(t13) → ε, ζ, η
  85. Text("The napoleon point will be within the original triangle when all angles are less than 150 degrees.") → text7
  86. Text("when an angle equals 150 degrees, then the napoleon point will equal that vertex.") → text8