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Overview of Special Lines in a Triangle

Overview of Special Lines in a Triangle
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2020-01-22 00:04
TriangleCoordinate GeometryGeometric Construction
This example demonstrates the properties and relationships of special lines in a triangle, such as angle bisectors, medians, and altitudes. It involves core mathematical concepts including vectors, coordinate geometry, inequalities, and geometric constructions. The construction utilizes commands and objects like Segment, Vector, Translate, AngularBisector, Intersect, OrthogonalLine, Circle, Midpoint, Polygon, Angle, Line, and Center. It is suitable for high school mathematics teaching and geometric construction learning.

GeoGebra Commands & Parameters

  1. Segment(B, A) → a
  2. Segment(A, C) → b
  3. Segment(C, B) → e
  4. Vector(C, H) → v
  5. Vector(C, J) → w
  6. Vector(C, I) → z
  7. Vector(C, C_2) → u
  8. Translate(B, u) → K
  9. Translate(C, u) → L
  10. Translate(A, u) → M
  11. Translate(B, u) → E'
  12. Translate(A, u) → G'
  13. Segment(E', G') → f
  14. Translate(A, u) → G'_1
  15. Translate(C, u) → F'
  16. Segment(G'_1, F') → g
  17. Translate(C, u) → F'_1
  18. Translate(B, u) → E'_1
  19. Segment(F'_1, E'_1) → h
  20. Translate(B, w) → D
  21. Translate(C, w) → N
  22. Translate(A, w) → O
  23. Translate(B, v) → P
  24. Translate(C, v) → Q
  25. Translate(A, v) → R
  26. Translate(C, v) → F'_2
  27. Translate(B, v) → E'_2
  28. Segment(F'_2, E'_2) → d
  29. Translate(B, v) → E'_3
  30. Translate(A, v) → G'_2
  31. Segment(E'_3, G'_2) → i
  32. Translate(A, v) → G'_3
  33. Translate(C, v) → F'_3
  34. Segment(G'_3, F'_3) → j
  35. Translate(C, w) → F'_4
  36. Translate(B, w) → E'_4
  37. Segment(F'_4, E'_4) → k
  38. Translate(B, w) → E'_5
  39. Translate(A, w) → G'_4
  40. Segment(E'_5, G'_4) → l
  41. Translate(A, w) → G'_5
  42. Translate(C, w) → F'_5
  43. Segment(G'_5, F'_5) → m
  44. Translate(B, z) → S
  45. Translate(B, z) → E'_6
  46. Translate(A, z) → G'_6
  47. Segment(E'_6, G'_6) → n
  48. Translate(A, z) → T
  49. Translate(C, z) → U
  50. Translate(C, z) → F'_6
  51. Translate(B, z) → E'_7
  52. Segment(F'_6, E'_7) → p
  53. Translate(A, z) → G'_7
  54. Translate(C, z) → F'_7
  55. Segment(G'_7, F'_7) → q
  56. AngularBisector(U, S, G'_6) → r
  57. AngularBisector(T, U, S) → s
  58. AngularBisector(S, T, U) → t
  59. Intersect(p, t) → V
  60. Intersect(n, s) → W
  61. Intersect(q, r) → Z
  62. Intersect(r, s) → A1
  63. Segment(S, Z) → a1
  64. Segment(U, W) → b1
  65. Segment(T, V) → c1
  66. OrthogonalLine(A1, q) → d1
  67. Intersect(q, d1) → B1
  68. Circle(A1, B1) → e1
  69. Midpoint(k) → M1
  70. Midpoint(l) → N1
  71. Midpoint(m) → O1
  72. Segment(D, O1) → l1
  73. Segment(N, N1) → m1
  74. Segment(O, M1) → n1
  75. Intersect(l1, m1) → P1
  76. Polygon(B, A, C) → poly1, f_1, e_1, g_1
  77. Angle(poly1) → α, β, γ
  78. Line(K, M) → a_1
  79. Line(M, L) → b_1
  80. Line(L, K) → c_1
  81. OrthogonalLine(K, b_1) → d_1
  82. OrthogonalLine(L, a_1) → h_1
  83. OrthogonalLine(M, c_1) → i_1
  84. Intersect(d_1, h_1) → D_1
  85. Circle(P, Q, R) → k_1
  86. Center(k_1) → E_1
  87. OrthogonalLine(E_1, i) → j_1
  88. OrthogonalLine(E_1, d) → l_1
  89. OrthogonalLine(E_1, j) → m_1
  90. Intersect(d, l_1) → F_1
  91. Intersect(i, j_1) → G_1
  92. Intersect(j, m_1) → H_1
  93. Segment(F_1, E_1) → n_1
  94. Segment(E_1, H_1) → p_1
  95. Segment(E_1, G_1) → q_1
  96. OrthogonalLine(D_1, b_1) → r_1
  97. OrthogonalLine(D_1, a_1) → s_1
  98. OrthogonalLine(D_1, c_1) → t_1
  99. Intersect(t_1, c_1) → A_1
  100. Intersect(r_1, b_1) → B_1
  101. Intersect(s_1, a_1) → C_1
  102. Segment(D_1, C_1) → a_2
  103. Segment(D_1, B_1) → b_2
  104. Segment(D_1, A_1) → c_2
  105. Segment(K, D_1) → d_2
  106. Segment(L, D_1) → e_2
  107. Segment(M, D_1) → f_2
  108. Segment(L, B_1) → g_2
  109. Segment(L, A_1) → h_2
  110. Segment(M, C_1) → i_2
  111. Translate(B, v) → E'_8
  112. Translate(A, v) → G'_8
  113. Translate(C, v) → F'_8
  114. Polygon(E'_8, G'_8, F'_8) → poly1', f'_8, e'_8, g'_8
  115. Translate(B, z) → E'_9
  116. Translate(A, z) → G'_9
  117. Translate(C, z) → F'_9
  118. Polygon(E'_9, G'_9, F'_9) → poly1'_1, f'_9, e'_9, g'_9
  119. Translate(B, w) → E'_{10}
  120. Translate(A, w) → G'_{10}
  121. Translate(C, w) → F'_{10}
  122. Polygon(E'_{10}, G'_{10}, F'_{10}) → poly1'_2, f'_{10}, e'_{10}, g'_{10}
  123. Translate(B, u) → E'_{11}
  124. Translate(A, u) → G'_{11}
  125. Translate(C, u) → F'_{11}
  126. Polygon(E'_{11}, G'_{11}, F'_{11}) → poly1'_3, f'_{11}, e'_{11}, g'_{11}
  127. Angle(K, A_1, D_1) → δ
  128. Angle(K, C_1, D_1) → ε
  129. Angle(D_1, B_1, M) → ζ
  130. Angle(R, G_1, E_1) → η
  131. Angle(E_1, H_1, R) → θ
  132. Angle(E_1, F_1, Q) → ι
  133. Segment(R, G_1) → j_2
  134. Segment(G_1, P) → k_2
  135. Segment(P, F_1) → l_2
  136. Segment(F_1, Q) → m_2
  137. Segment(Q, H_1) → n_2
  138. Segment(H_1, R) → p_2
  139. Segment(N, M1) → q_2
  140. Segment(M1, D) → r_2
  141. Segment(D, N1) → s_2
  142. Segment(N1, O) → t_2
  143. Segment(O, O1) → a_3
  144. Segment(O1, N) → b_3
  145. Angle(Z, T, A1) → κ
  146. Angle(A1, T, W) → λ
  147. Angle(A1, U, T) → μ
  148. Angle(S, U, A1) → ν
  149. Angle(A1, S, U) → ξ
  150. Angle(T, S, Z) → ο
  151. Line(H_1, E_1) → c
  152. Line(G_1, E_1) → c_3
  153. Line(F_1, E_1) → d_3
  154. Intersect(k_1, d_3) → E, F
  155. Intersect(k_1, c_3) → G, I_1
  156. Intersect(k_1, c) → J_1, K_1
  157. Segment(G, I_1) → e_3
  158. Segment(J_1, K_1) → f_3
  159. Segment(F, E) → g_3
  160. Circle(A, C, B) → h_3
  161. Midpoint(g_1) → L_1
  162. Midpoint(f_1) → M_1
  163. Segment(M_1, C) → i_3
  164. Segment(L_1, A) → j_3
  165. Intersect(i_3, j_3) → N_1
  166. Midpoint(C_2, N_1) → O_1
  167. Midpoint(N_1, I) → Q_1
  168. Midpoint(N_1, J) → R_1
  169. Midpoint(N_1, O_1) → T_1
  170. Midpoint(R_1, J) → U_1
  171. Midpoint(Q_1, I) → V_1
  172. Vector(T_1, O_1) → u_1
  173. Vector(Q_1, V_1) → v_1
  174. Midpoint(U_1, J) → P_1
  175. Midpoint(N_1, H) → S_1
  176. Midpoint(S_1, L_1) → W_1
  177. Midpoint(S_1, W_1) → Z_1
  178. Midpoint(S_1, Z_1) → A_2
  179. Vector(W_1, A_2) → w_1
  180. Vector(R_1, P_1) → z_1