MathPlotAI Logo MathPlotAI

Heights, Medians, and Angle Bisectors of Triangles

Heights, Medians, and Angle Bisectors of Triangles
Loading the math board and drawing, please wait…
2014-01-14 03:05
TriangleAngleGeometric ConstructionSlider AnimationLocus
① This interactive GeoGebra example dynamically illustrates the three key cevians of a triangle: altitudes (heights), medians, and angle bisectors, showing their intersections and relationships. ② Core mathematical concepts include the orthocenter, centroid, and incenter, along with the angle bisector theorem, median properties, and symmetry in triangles. ③ Construction details: The applet uses GeoGebra commands such as Segment, Angle, Circle, AngularBisector, Intersect, OrthogonalLine, Point, Midpoint, Rotate, and Polygon to build the triangle and its auxiliary lines. Multiple labeled objects (e.g., D, E, G, d, H, I, etc.) are used to mark vertices, feet, midpoints, and angle bisector points, enabling dynamic manipulation. ④ Suitable for middle and high school geometry classes (grades 8–12), this resource is ideal for teaching triangle properties, geometric construction, and exploratory learning, whether in classroom demonstrations or student self-study.

GeoGebra Commands & Parameters

  1. Segment(E, D) → d
  2. Segment(G, H) → g
  3. Segment(H, I) → h
  4. Segment(G, I) → i
  5. Segment(E, F) → e
  6. Segment(D, F) → f
  7. Angle(D, E, F) → δ
  8. Angle(E, F, D) → ε
  9. Angle(F, D, E) → ζ
  10. Angle(G, I, H) → η
  11. Angle(I, H, G) → θ
  12. Angle(H, G, I) → ι
  13. Circle(E, D, F) → p
  14. Circle(G, H, I) → q
  15. AngularBisector(F, E, D) → b_1
  16. AngularBisector(E, D, F) → a_1
  17. AngularBisector(D, F, E) → c_1
  18. Intersect(b_1, f) → N
  19. Segment(E, N) → d_1
  20. Intersect(a_1, e) → O
  21. Segment(D, O) → e_1
  22. Intersect(c_1, d) → P
  23. Segment(P, F) → f_1
  24. AngularBisector(G, H, I) → g_1
  25. AngularBisector(I, G, H) → h_1
  26. AngularBisector(H, I, G) → i_1
  27. Intersect(g_1, i) → Q
  28. Segment(H, Q) → j_1
  29. Intersect(i_1, g) → R
  30. Segment(I, R) → k_1
  31. Intersect(h_1, h) → S
  32. Segment(G, S) → l_1
  33. Segment(T, U) → m_1
  34. Segment(U, V) → n_1
  35. Segment(V, T) → p_1
  36. Segment(W, Z) → q_1
  37. Segment(Z, A_1) → r_1
  38. Segment(W, A_1) → s_1
  39. Circle(T, U, V) → t_1
  40. Circle(W, Z, A_1) → c_2
  41. Segment(B_1, C_1) → a_2
  42. OrthogonalLine(C_1, a_2) → b_2
  43. Point(b_2) → E_1
  44. Segment(B_1, E_1) → d_2
  45. Segment(C_1, E_1) → f_2
  46. Circle(B_1, C_1, E_1) → e_2
  47. Midpoint(a_2) → D_1
  48. Midpoint(d_2) → F_1
  49. Midpoint(f_2) → G_1
  50. Midpoint(r_1) → H_1
  51. Midpoint(q_1) → I_1
  52. Midpoint(s_1) → J_1
  53. Midpoint(m_1) → K_1
  54. Midpoint(p_1) → L_1
  55. Midpoint(n_1) → M_1
  56. Segment(U, L_1) → g_2
  57. Segment(K_1, V) → h_2
  58. Segment(M_1, T) → i_2
  59. Segment(I_1, A_1) → j_2
  60. Segment(W, H_1) → k_2
  61. Segment(Z, J_1) → l_2
  62. Segment(B_1, G_1) → m_2
  63. Segment(C_1, F_1) → n_2
  64. Segment(D_1, E_1) → p_2
  65. Intersect(g_2, h_2) → N_1
  66. Intersect(j_2, k_2) → O_1
  67. Intersect(m_2, n_2) → P_1
  68. Segment(Q_1, R_1) → q_2
  69. Segment(T_1, U_1) → t_2
  70. Segment(T_1, V_1) → a_3
  71. Segment(U_1, V_1) → b_3
  72. Segment(Z_1, A_2) → d_3
  73. Rotate(A_2, 90*°, Z_1) → B_2
  74. Angle(A_2, Z_1, B_2) → κ
  75. Segment(B_2, Z_1) → e_3
  76. Segment(B_2, A_2) → c_3
  77. Circle(T_1, U_1, V_1) → g_3
  78. Circle(B_2, Z_1, A_2) → h_3
  79. OrthogonalLine(S_1, q_2) → i_3
  80. OrthogonalLine(U_1, a_3) → p_3
  81. OrthogonalLine(T_1, b_3) → q_3
  82. OrthogonalLine(V_1, t_2) → r_3
  83. Intersect(p_3, a_3) → F_2
  84. Segment(U_1, F_2) → s_3
  85. OrthogonalLine(T_1, b_3) → t_3
  86. OrthogonalLine(V_1, t_2) → a_4
  87. Intersect(t_3, a_4) → E_2
  88. Segment(E_2, F_2) → b_4
  89. Segment(T_1, E_2) → c_4
  90. Segment(E_2, V_1) → d_4
  91. Point(d_4) → G_2
  92. Segment(T_1, G_2) → e_4
  93. Point(e_4) → H_2
  94. OrthogonalLine(V_1, c_4) → g_4
  95. OrthogonalLine(T_1, d_4) → h_4
  96. Segment(T_1, G_2) → i_4
  97. Intersect(g_4, c_4) → I_2
  98. Segment(I_2, V_1) → j_4
  99. Segment(U_1, G_2) → k_4
  100. Segment(I_2, U_1) → l_4
  101. Segment(Q_1, W_1) → r_2
  102. Segment(R_1, W_1) → s_2
  103. Circle(Q_1, R_1, W_1) → k
  104. OrthogonalLine(W_1, q_2) → a
  105. OrthogonalLine(Q_1, s_2) → b
  106. OrthogonalLine(R_1, r_2) → j
  107. Intersect(k, j, 2) → A
  108. Intersect(k, a, 2) → J
  109. Intersect(k, b, 2) → K
  110. Segment(R_1, A) → l
  111. Segment(J, W_1) → m
  112. Segment(K, Q_1) → n
  113. Intersect(r_2, l) → L
  114. Intersect(s_2, n) → M
  115. Intersect(q_2, m) → C_2
  116. Polygon(R_1, Q_1, W_1) → poly1, w_1, r_4, q_4
  117. Polygon(T_1, U_1, V_1) → poly2, v_1, t_4, u_1
  118. Segment(T_1, U_1) → r
  119. Segment(D_2, J_2) → s
  120. OrthogonalLine(D_2, s) → t
  121. Point(t) → K_2
  122. Segment(K_2, J_2) → f_3
  123. Segment(K_2, D_2) → j_3
  124. Circle(K_2, D_2, J_2) → k_3
  125. Angle(D_2, K_2, J_2) → α
  126. Angle(K_2, J_2, D_2) → β
  127. Angle(J_2, D_2, K_2) → γ
  128. Angle(G, I, R) → λ
  129. OrthogonalLine(Z_1, c_3) → c
  130. OrthogonalLine(B_2, d_3) → l_3
  131. OrthogonalLine(A_2, e_3) → m_3
  132. Intersect(c, c_3) → B
  133. Segment(Z_1, B) → n_3
  134. Segment(B_2, Z_1) → f_4
  135. Segment(Z_1, A_2) → m_4
  136. AngularBisector(j_3, f_3) → n_4, p_4
  137. AngularBisector(f_3, j_3) → s_4, a_5
  138. AngularBisector(s, j_3) → b_5, c_5
  139. AngularBisector(f_3, s) → d_5, e_5
  140. Intersect(d_5, j_3) → C
  141. Intersect(c_5, f_3) → L_2
  142. Point(n_4) → M_2
  143. Segment(K_2, M_2) → f_5
  144. Segment(L_2, D_2) → g_5
  145. Segment(C, J_2) → h_5
  146. Intersect(j_1, k_1) → N_2
  147. Intersect(d_1, e_1) → O_2
  148. Intersect(f_5, g_5) → P_2
  149. Polygon(B_2, Z_1, A_2) → poly3, a_6, b_6, z_1
  150. Polygon(T, U, V) → poly4, v, t_5, u
  151. Polygon(W, Z, A_1) → poly5, a_7, w, z
  152. Polygon(B_1, C_1, E_1) → poly6, e_6, b_7, c_6
  153. Polygon(I, G, H) → poly7, h_6, ı, g_6
  154. Polygon(E, D, F) → poly8, f_6, e_7, d_6
  155. Polygon(K_2, D_2, J_2) → poly9, j_5, k_5, d_7

AI Drawing Prompt Reference

Draw a triangle ABC, connect each vertex to the midpoint of the opposite side to form three medians, connect each vertex to the foot of the perpendicular from the opposite side to form three altitudes, and connect each vertex to the intersection of the angle bisectors of the opposite side to form three angle bisectors. Use rotation and intersection operations to demonstrate the triangle's centroid, orthocenter, and incenter.
The prompt above was compiled from the original drawing commands and is for reference only.