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Animation Demonstrating Circle, Orthocenter, and Polygon Relations

Animation Demonstrating Circle, Orthocenter, and Polygon Relations
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2016-06-20 09:16
TriangleCirclePolygonCoordinate GeometrySolid Geometry
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 relationship between circumcircle, orthocenter, special points, and secant line through animation. Key math concepts include circumcircle, orthocenter, polygons, inequalities, cylinder, centroid, and secant line. Construction involves commands like Line, Segment, Midpoint, OrthogonalLine, Intersect, Circle, Dilate, Rotate, Polygon, Point, Ray, CircleSector, Angle, CircleArc, Vector, and Translate. Suitable for high school and above, focusing on solid geometry and coordinate geometry applications.

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