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Triangle Centroid, Orthocenter, and Special Points Animation

Triangle Centroid, Orthocenter, and Special Points Animation
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2016-06-26 09:32
TriangleVectorTrigonometryCoordinate GeometryGeometric Construction
In collections CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2
This example demonstrates the dynamic behavior and relationships between the centroid, orthocenter, and other special points of a triangle through animation. Key math concepts include centroid, orthocenter, special points, coordinate geometry, and geometric construction. It utilizes commands like Line, Segment, Midpoint, OrthogonalLine, Intersect, Min, Circle, Dilate, Rotate, Polygon, Point, Ray, CircleSector, Angle, CircleArc, Vector, and Translate in GeoGebra for construction. Suitable for high school mathematics education to understand the properties and interconnections of triangle special points.

GeoGebra Commands & Parameters

  1. Line(C, B) → l_1
  2. Segment(B, C) → g
  3. Midpoint(g) → F_3
  4. OrthogonalLine(F_3, l_1) → j_3
  5. Line(A, C) → f_1
  6. Segment(C, A) → h
  7. Midpoint(h) → E_3
  8. OrthogonalLine(E_3, f_1) → k_3
  9. Intersect(k_3, j_3) → I_2
  10. Midpoint(I_2, C) → I_3
  11. Segment(A, B) → f
  12. Midpoint(f) → D_3
  13. Min(list1) → a
  14. Circle(D_3, (0.15 * a)) → c
  15. Circle(E_3, (0.15 * a)) → d
  16. Circle(F_3, (0.15 * a)) → e
  17. Intersect(c, f, 2) → H
  18. Intersect(d, h, 2) → K
  19. Intersect(e, g, 2) → M
  20. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], D_3) → A'
  21. Dilate(B, If[2 < b ≤ 3, 3 - b, 0], F_3) → B'
  22. Dilate(C, If[4 < b ≤ 5, 5 - b, 0], E_3) → C'
  23. Segment(A, A') → l
  24. Segment(B, B') → m
  25. Segment(C, C') → n
  26. Rotate(A, If[1 < b ≤ 2, ((b - 1) * π), π], D_3) → A'_1
  27. Rotate(A', If[1 < b ≤ 2, ((b - 1) * π), π], D_3) → A''
  28. Segment(A'_1, A'') → l'
  29. Rotate(B, If[3 < b ≤ 4, ((b - 3) * π), π], F_3) → B'_1
  30. Rotate(B', If[3 < b ≤ 4, ((b - 3) * π), π], F_3) → B''
  31. Segment(B'_1, B'') → m'
  32. Rotate(C, If[5 < b ≤ 6, ((b - 5) * π), π], E_3) → C'_1
  33. Rotate(C', If[5 < b ≤ 6, ((b - 5) * π), π], E_3) → C''
  34. Segment(C'_1, C'') → n'
  35. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  36. Line(A, B) → i
  37. Point(i) → C_3
  38. Point(i) → G
  39. OrthogonalLine(C, i) → j
  40. Intersect(i, j) → I
  41. Ray(A, G) → k
  42. Ray(B, C_3) → p
  43. Point(k) → J
  44. Point(p) → L
  45. Circle(I, (0.15 * a)) → s
  46. Intersect(s, i, 2) → N
  47. Intersect(s, j, 2) → O_3
  48. CircleSector(I, N, O_3) → t
  49. Angle(C, B, A) → α
  50. Angle(B, A, C) → β
  51. Angle(A, C, B) → γ
  52. Point(f_1) → P
  53. Point(f_1) → Q
  54. Ray(A, P) → g_1
  55. Ray(C, Q) → h_1
  56. Point(g_1) → R
  57. Point(h_1) → S
  58. OrthogonalLine(B, f_1) → k_1
  59. Intersect(f_1, k_1) → T
  60. Circle(T, (0.15 * a)) → d_1
  61. Intersect(d_1, f_1, 1) → U
  62. Intersect(d_1, k_1, 1) → V
  63. CircleSector(T, U, V) → e_1
  64. OrthogonalLine(A, l_1) → m_1
  65. Intersect(l_1, m_1) → W
  66. Circle(W, (0.15 * a)) → p_1
  67. Intersect(p_1, l_1, 1) → Z
  68. Intersect(p_1, m_1, 1) → A_1
  69. Point(l_1) → B_1
  70. Point(l_1) → D_1
  71. CircleSector(W, Z, A_1) → q_1
  72. Ray(C, B_1) → n_1
  73. Ray(B, D_1) → r_1
  74. Point(n_1) → E_1
  75. Point(r_1) → F_1
  76. Dilate(C, If[6 < b ≤ 7, 7 - b, 0], I) → G_1
  77. Segment(C, G_1) → f_2
  78. Dilate(B, If[7 < b ≤ 8, 8 - b, 0], T) → B'_2
  79. Segment(B, B'_2) → g_2
  80. Dilate(A, If[8 < b ≤ 9, 9 - b, 0], W) → A'_2
  81. Segment(A, A'_2) → h_2
  82. Intersect(k_1, m_1) → O
  83. Dilate(I, If[12 < b ≤ 13, 13 - b, 0], O) → I'
  84. Dilate(A, If[12 < b ≤ 13, 13 - b, 0], O) → A'_3
  85. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], O) → T'
  86. Segment(I, I') → i_2
  87. Segment(A, A'_3) → j_2
  88. Segment(T, T') → k_2
  89. Dilate(C, If[12 < b ≤ 13, 13 - b, 0], O) → I_1
  90. Dilate(W, If[12 < b ≤ 13, 13 - b, 0], O) → W'
  91. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], O) → T'_1
  92. Segment(W, W') → l_2
  93. Segment(C, I_1) → m_2
  94. Segment(T, T'_1) → n_2
  95. Dilate(I, If[12 < b ≤ 13, 13 - b, 0], O) → I'_1
  96. Dilate(B, If[12 < b ≤ 13, 13 - b, 0], O) → B'_3
  97. Dilate(W, If[12 < b ≤ 13, 13 - b, 0], O) → W'_1
  98. Segment(B, B'_3) → p_2
  99. Segment(W, W'_1) → q_2
  100. Segment(I, I'_1) → r_2
  101. Midpoint(O, C) → F
  102. Midpoint(O, A) → D
  103. Midpoint(O, B) → E
  104. Dilate(O, If[14 < b ≤ 15, 15 - b, 0], F) → M_1
  105. Dilate(F, If[15 < b ≤ 16, 16 - b, 0], C) → N_1
  106. Segment(F, N_1) → t_2
  107. Rotate(O, If[16 < b ≤ 17, ((b - 16) * π), π], F) → O_1
  108. Rotate(M_1, If[16 < b ≤ 17, ((b - 16) * π), π], F) → P_1
  109. Dilate(O, If[17 < b ≤ 18, 18 - b, 0], E) → Q_1
  110. Dilate(E, If[18 < b ≤ 19, 19 - b, 0], B) → R_1
  111. Segment(E, R_1) → e_2
  112. Rotate(O, If[19 < b ≤ 20, ((b - 19) * π), π], E) → S_1
  113. Rotate(Q_1, If[19 < b ≤ 20, ((b - 19) * π), π], E) → T_1
  114. Dilate(O, If[20 < b ≤ 21, 21 - b, 0], D) → U_1
  115. Dilate(D, If[21 < b ≤ 22, 22 - b, 0], A) → V_1
  116. Segment(D, V_1) → h_3
  117. Rotate(O, If[22 < b ≤ 23, ((b - 22) * π), π], D) → W_1
  118. Rotate(U_1, If[22 < b ≤ 23, ((b - 22) * π), π], D) → Z_1
  119. Intersect(e, l_1) → A_2, B_2
  120. Intersect(e, j_3) → C_2, D_2
  121. CircleSector(F_3, A_2, C_2) → c_3
  122. Circle(E_3, (0.15 * a)) → d_3
  123. Intersect(d_3, k_3) → E_2, F_2
  124. Intersect(d_3, f_1) → G_2, H_2
  125. CircleSector(E_3, G_2, E_2) → e_3
  126. Midpoint(O, I_2) → J_2
  127. Dilate(O, If[28 < b ≤ 29, 29 - b, 0], J_2) → K_2
  128. Dilate(J_2, If[29 < b ≤ 30, 30 - b, 0], I_2) → L_2
  129. Rotate(O, If[30 < b ≤ 31, ((b - 30) * π), π], J_2) → M_2
  130. Rotate(K_2, If[30 < b ≤ 31, ((b - 30) * π), π], J_2) → N_2
  131. Dilate(J_2, If[31 < b ≤ 32, 32 - b, 0], D_3) → O_2
  132. Rotate(J_2, If[32 < b ≤ 34, ((b - 32) * π), (2 * π)], J_2) → P_2
  133. Rotate(O_2, If[32 < b ≤ 34, ((b - 32) * π), (2 * π)], J_2) → Q_2
  134. Segment(P_2, Q_2) → s_3
  135. CircleArc(J_2, D_3, Q_2) → t_3
  136. Circle(J_2, D_3) → c_4
  137. Point(j) → R_2
  138. Point(j) → S_2
  139. Ray(C, R_2) → j_4
  140. Ray(I, S_2) → k_4
  141. Point(j_4) → T_2
  142. Point(k_4) → U_2
  143. Angle(B, T, C) → δ
  144. Angle(C, I, A) → ε
  145. Angle(A, W, B) → ζ
  146. Angle(I_2, E_3, C) → η
  147. Angle(I_2, D_3, A) → θ
  148. Angle(I_2, F_3, B) → ι
  149. Line(I_2, D_3) → s_2
  150. Line(A, W) → a_2
  151. Line(C, I) → b_2
  152. Line(B, T) → c_2
  153. Dilate(I_2, If[0 < v ≤ 1, 1 - v, 0], J_2) → C_1
  154. Dilate(J_2, If[1 < v ≤ 2, 2 - v, 0], O) → H_1
  155. Segment(J_2, H_1) → f_3
  156. Rotate(I_2, If[2 < v ≤ 3, ((v - 2) * π), π], J_2) → K_1
  157. Rotate(C_1, If[2 < v ≤ 3, ((v - 2) * π), π], J_2) → V_2
  158. Dilate(J_2, If[3 < v ≤ 4, 4 - v, 0], D_3) → W_2
  159. Segment(J_2, W_2) → i_3
  160. Rotate(J_2, If[4 < v ≤ 6, ((v - 4) * π), (2 * π)], J_2) → Z_2
  161. Rotate(W_2, If[4 < v ≤ 6, ((v - 4) * π), (2 * π)], J_2) → A_3
  162. CircleArc(J_2, D_3, A_3) → p_3
  163. Dilate(I_2, If[6 < v ≤ 7, 7 - v, 0], I_3) → L_1
  164. Dilate(I_3, If[7 < v ≤ 8, 8 - v, 0], C) → B_3
  165. Segment(I_3, B_3) → n_3
  166. Dilate(J_2, If[8 < v ≤ 9, 9 - v, 0], I_2) → G_3
  167. Vector(J_2, G_3) → w_1
  168. Translate(Z_2, w_1) → H_3
  169. Translate(A_3, w_1) → J_3
  170. Segment(H_3, J_3) → q_3
  171. Rotate(H_3, If[9 < v ≤ 10, ((v - 9) * π), π], I_2) → K_3
  172. Rotate(J_3, If[9 < v ≤ 10, ((v - 9) * π), π], I_2) → L_3
  173. Segment(K_3, L_3) → r_3
  174. Rotate(K_3, If[10 < v ≤ 11, ((v - 10) * π), π], I_3) → M_3
  175. Rotate(L_3, If[10 < v ≤ 11, ((v - 10) * π), π], I_3) → N_3
  176. Segment(M_3, N_3) → g_4
  177. Midpoint(I_2, A) → R_3
  178. Midpoint(I_2, B) → S_3
  179. Dilate(I_2, If[6 < v ≤ 7, 7 - v, 0], R_3) → U_3
  180. Dilate(I_2, If[6 < v ≤ 7, 7 - v, 0], S_3) → W_3
  181. Dilate(R_3, If[7 < v ≤ 8, 8 - v, 0], A) → J_1
  182. Dilate(S_3, If[7 < v ≤ 8, 8 - v, 0], B) → B_4
  183. Segment(R_3, J_1) → f_4
  184. Segment(S_3, B_4) → h_4
  185. Dilate(J_2, If[11 < v ≤ 12, 12 - v, 0], I_2) → D_4
  186. Vector(J_2, D_4) → u_1
  187. Translate(Z_2, u_1) → G_4
  188. Translate(A_3, u_1) → H_4
  189. Segment(G_4, H_4) → l_4
  190. Angle(D, J_2, D_3) → μ
  191. Rotate(G_4, If[12 < v ≤ 13, ((-(v - 12)) * μ), (-μ)], I_2) → P_3
  192. Rotate(H_4, If[12 < v ≤ 13, ((-(v - 12)) * μ), (-μ)], I_2) → Q_3
  193. Segment(P_3, Q_3) → m_4
  194. Rotate(P_3, If[13 < v ≤ 14, ((-(v - 13)) * π), (-π)], R_3) → T_3
  195. Rotate(Q_3, If[13 < v ≤ 14, ((-(v - 13)) * π), (-π)], R_3) → V_3
  196. Segment(T_3, V_3) → n_4
  197. Dilate(J_2, If[14 < v ≤ 15, 15 - v, 0], I_2) → Z_3
  198. Vector(J_2, Z_3) → v_1
  199. Translate(Z_2, v_1) → A_4
  200. Translate(A_3, v_1) → C_4
  201. Segment(A_4, C_4) → p_4
  202. Angle(D_3, J_2, E) → κ
  203. Rotate(A_4, If[15 < v ≤ 16, ((v - 15) * κ), κ], I_2) → E_4
  204. Rotate(C_4, If[15 < v ≤ 16, ((v - 15) * κ), κ], I_2) → F_4
  205. Segment(E_4, F_4) → r_4
  206. Rotate(E_4, If[16 < v ≤ 17, ((v - 16) * π), π], S_3) → I_4
  207. Rotate(F_4, If[16 < v ≤ 17, ((v - 16) * π), π], S_3) → J_4
  208. Segment(I_4, J_4) → s_4
  209. Circle(I_2, C) → d_4
  210. Rotate(B, If[17 < v ≤ 19, ((v - 17) * π), (2 * π)], I_2) → B'_4
  211. CircleArc(I_2, B, B'_4) → e_4
  212. Line(B, C) → q
  213. Line(A, C) → r
  214. Line(B, A) → i_1
  215. Segment(A, F_3) → j_1
  216. Segment(B, E_3) → s_1
  217. Segment(C, D_3) → t_1
  218. Intersect(t_1, s_1) → K_4
  219. Segment(K_4, J_2) → d_2
  220. Line(O, I_2) → g_3
  221. Circle(J_2, d_2) → q_4
  222. Intersect(q_4, g_3, 1) → L_4
  223. Circle(L_4, d_2) → t_4
  224. Intersect(t_4, g_3, 1) → M_4
  225. Dilate(K_4, If[0 < w ≤ 1, 1 - w, 0], J_2) → N_4
  226. Segment(K_4, N_4) → l_3
  227. Dilate(J_2, If[1 < w ≤ 2, 2 - w, 0], L_4) → O_4
  228. Segment(J_2, O_4) → m_3
  229. Dilate(L_4, If[2 < w ≤ 3, 3 - w, 0], M_4) → P_4
  230. Segment(L_4, P_4) → a_3
  231. Dilate(M_4, If[3 < w ≤ 4, 4 - w, 0], O) → Q_4
  232. Segment(M_4, Q_4) → b_3
  233. Rotate(K_4, If[4 < w ≤ 5, ((w - 4) * π), π], J_2) → R_4
  234. Rotate(N_4, If[4 < w ≤ 5, ((w - 4) * π), π], J_2) → S_4
  235. Segment(R_4, S_4) → i_4
  236. Rotate(R_4, If[5 < w ≤ 6, ((w - 5) * π), π], L_4) → T_4
  237. Rotate(S_4, If[5 < w ≤ 6, ((w - 5) * π), π], L_4) → U_4
  238. Segment(T_4, U_4) → a_4
  239. Rotate(T_4, If[6 < w ≤ 7, ((w - 6) * π), π], M_4) → V_4
  240. Rotate(U_4, If[6 < w ≤ 7, ((w - 6) * π), π], M_4) → W_4
  241. Segment(V_4, W_4) → b_4
  242. Line(O, I_2) → f_5

AI Drawing Prompt Reference

Draw a triangle and label its centroid, orthocenter, and related special points, demonstrating the dynamic relationships between them through animation. Start by drawing the triangle's three sides, then find the midpoints of each side and draw perpendicular lines to these sides—the intersection of these lines is the orthocenter. Next, connect each vertex to the midpoint of the opposite side to find the centroid. Draw circles centered at these midpoints with a certain radius, and mark the intersections of these circles with the sides as special points. Finally, rotate and scale these points to show their dynamic behavior.
The prompt above was compiled from the original drawing commands and is for reference only.