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9-Point Circle Construction and Properties

9-Point Circle Construction and Properties
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2016-09-14 09:33
TriangleCircleCoordinate GeometryGeometric Construction
This example demonstrates how to construct the nine-point circle of a triangle by plotting the midpoints of the three altitudes, the midpoints of each side, and the midpoints of the segments connecting each vertex to the orthocenter. It illustrates the relationship between the nine-point circle and the circumcircle of the triangle. Key mathematical concepts include triangle, orthocenter, midpoint, circle, geometric construction, and limits. The construction uses GeoGebra commands such as Segment, Midpoint, Circle, Intersect, Dilate, Rotate, Polygon, Line, Point, OrthogonalLine, Ray, CircleSector, and Angle. This is suitable for high school and above students to understand the construction method and applications of the nine-point circle in geometric proofs.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Segment(B, C_1) → g
  3. Segment(C_1, A) → h
  4. Midpoint(f) → D
  5. Midpoint(h) → E
  6. Midpoint(g) → F
  7. Min(list1) → a
  8. Circle(D, (0.1 * a)) → c
  9. Circle(E, (0.1 * a)) → d
  10. Circle(F, (0.1 * a)) → e
  11. Intersect(c, f, 2) → H
  12. Intersect(d, h, 2) → K
  13. Intersect(e, g, 2) → M
  14. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], D) → A'
  15. Dilate(B, If[2 < b ≤ 3, 3 - b, 0], F) → B'
  16. Dilate(C_1, If[4 < b ≤ 5, 5 - b, 0], E) → C'
  17. Segment(A, A') → l
  18. Segment(B, B') → m
  19. Segment(C_1, C') → n
  20. Rotate(A, If[1 < b ≤ 2, ((b - 1) * π), π], D) → A'_1
  21. Rotate(A', If[1 < b ≤ 2, ((b - 1) * π), π], D) → A''
  22. Segment(A'_1, A'') → l'
  23. Rotate(B, If[3 < b ≤ 4, ((b - 3) * π), π], F) → B'_1
  24. Rotate(B', If[3 < b ≤ 4, ((b - 3) * π), π], F) → B''
  25. Segment(B'_1, B'') → m'
  26. Rotate(C_1, If[5 < b ≤ 6, ((b - 5) * π), π], E) → C'_1
  27. Rotate(C', If[5 < b ≤ 6, ((b - 5) * π), π], E) → C''
  28. Segment(C'_1, C'') → n'
  29. Polygon(A, B, C_1) → poly1, c_1, a_1, b_1
  30. Line(A, B) → i
  31. Point(i) → C
  32. Point(i) → G
  33. OrthogonalLine(C_1, i) → j
  34. Intersect(i, j) → I
  35. Ray(A, G) → k
  36. Ray(B, C) → p
  37. Point(k) → J
  38. Point(p) → L
  39. Ray(A, J) → q
  40. Ray(B, L) → r
  41. Circle(I, (0.1 * a)) → s
  42. Intersect(s, i, 2) → N
  43. Intersect(s, j, 2) → O
  44. CircleSector(I, N, O) → t
  45. Angle(C_1, B, A) → α
  46. Angle(B, A, C_1) → β
  47. Angle(A, C_1, B) → γ
  48. Line(A, C_1) → f_1
  49. Point(f_1) → P
  50. Point(f_1) → Q
  51. Ray(A, P) → g_1
  52. Ray(C_1, Q) → h_1
  53. Point(g_1) → R
  54. Ray(A, R) → i_1
  55. Point(h_1) → S
  56. Ray(C_1, S) → j_1
  57. OrthogonalLine(B, f_1) → k_1
  58. Intersect(f_1, k_1) → T
  59. Circle(T, (0.1 * 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. Line(C_1, B) → l_1
  64. OrthogonalLine(A, l_1) → m_1
  65. Intersect(l_1, m_1) → W
  66. Circle(W, (0.1 * 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_1, B_1) → n_1
  73. Ray(B, D_1) → r_1
  74. Point(n_1) → E_1
  75. Point(r_1) → F_1
  76. Ray(C_1, E_1) → s_1
  77. Ray(B, F_1) → t_1
  78. Dilate(C_1, If[6 < b ≤ 7, 7 - b, 0], I) → G_1
  79. Segment(C_1, G_1) → f_2
  80. Dilate(B, If[7 < b ≤ 8, 8 - b, 0], T) → B'_2
  81. Segment(B, B'_2) → g_2
  82. Dilate(A, If[8 < b ≤ 9, 9 - b, 0], W) → A'_2
  83. Segment(A, A'_2) → h_2
  84. Intersect(k_1, m_1) → H_1
  85. Rotate(t, If[9 < b ≤ 10, ((b - 9) * π / 2), π / 2], I) → t'
  86. Rotate(e_1, If[10 < b ≤ 11, ((b - 10) * π / 2), π / 2], T) → c_2
  87. Rotate(q_1, If[11 < b ≤ 12, ((b - 11) * π / 2), π / 2], W) → d_2
  88. Dilate(I, If[12 < b ≤ 13, 13 - b, 0], H_1) → I'
  89. Dilate(A, If[12 < b ≤ 13, 13 - b, 0], H_1) → A'_3
  90. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], H_1) → T'
  91. Segment(I, I') → i_2
  92. Segment(A, A'_3) → j_2
  93. Segment(T, T') → k_2
  94. Dilate(C_1, If[12 < b ≤ 13, 13 - b, 0], H_1) → I_1
  95. Dilate(W, If[12 < b ≤ 13, 13 - b, 0], H_1) → W'
  96. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], H_1) → T'_1
  97. Segment(W, W') → l_2
  98. Segment(C_1, I_1) → m_2
  99. Segment(T, T'_1) → n_2
  100. Dilate(I, If[12 < b ≤ 13, 13 - b, 0], H_1) → I'_1
  101. Dilate(B, If[12 < b ≤ 13, 13 - b, 0], H_1) → B'_3
  102. Dilate(W, If[12 < b ≤ 13, 13 - b, 0], H_1) → W'_1
  103. Segment(B, B'_3) → p_2
  104. Segment(W, W'_1) → q_2
  105. Segment(I, I'_1) → r_2
  106. Midpoint(H_1, C_1) → J_1
  107. Midpoint(H_1, A) → K_1
  108. Midpoint(H_1, B) → L_1
  109. Dilate(H_1, If[14 < b ≤ 15, 15 - b, 0], J_1) → M_1
  110. Segment(H_1, M_1) → s_2
  111. Dilate(J_1, If[15 < b ≤ 16, 16 - b, 0], C_1) → N_1
  112. Segment(J_1, N_1) → t_2
  113. Rotate(H_1, If[16 < b ≤ 17, ((b - 16) * π), π], J_1) → O_1
  114. Rotate(M_1, If[16 < b ≤ 17, ((b - 16) * π), π], J_1) → P_1
  115. Segment(O_1, P_1) → a_2
  116. Dilate(H_1, If[17 < b ≤ 18, 18 - b, 0], L_1) → Q_1
  117. Segment(H_1, Q_1) → b_2
  118. Dilate(L_1, If[18 < b ≤ 19, 19 - b, 0], B) → R_1
  119. Segment(L_1, R_1) → e_2
  120. Rotate(H_1, If[19 < b ≤ 20, ((b - 19) * π), π], L_1) → S_1
  121. Rotate(Q_1, If[19 < b ≤ 20, ((b - 19) * π), π], L_1) → T_1
  122. Segment(S_1, T_1) → f_3
  123. Dilate(H_1, If[20 < b ≤ 21, 21 - b, 0], K_1) → U_1
  124. Segment(H_1, U_1) → g_3
  125. Dilate(K_1, If[21 < b ≤ 22, 22 - b, 0], A) → V_1
  126. Segment(K_1, V_1) → h_3
  127. Rotate(H_1, If[22 < b ≤ 23, ((b - 22) * π), π], K_1) → W_1
  128. Rotate(U_1, If[22 < b ≤ 23, ((b - 22) * π), π], K_1) → Z_1
  129. Segment(W_1, Z_1) → i_3
  130. Intersect(e, l_1) → A_2, B_2
  131. OrthogonalLine(F, l_1) → j_3
  132. Intersect(e, j_3) → C_2, D_2
  133. CircleSector(F, A_2, C_2) → c_3
  134. OrthogonalLine(E, f_1) → k_3
  135. Circle(E, (0.1 * a)) → d_3
  136. Intersect(d_3, k_3) → E_2, F_2
  137. Intersect(d_3, f_1) → G_2, H_2
  138. CircleSector(E, G_2, E_2) → e_3
  139. Rotate(c_3, If[25 < b ≤ 26, ((b - 25) * π / 2), π / 2], F) → p_3
  140. Rotate(e_3, If[26 < b ≤ 27, ((b - 26) * π / 2), π / 2], E) → q_3
  141. Intersect(k_3, j_3) → I_2
  142. Midpoint(H_1, I_2) → J_2
  143. Dilate(H_1, If[28 < b ≤ 29, 29 - b, 0], J_2) → K_2
  144. Segment(H_1, K_2) → l_3
  145. Dilate(J_2, If[29 < b ≤ 30, 30 - b, 0], I_2) → L_2
  146. Segment(J_2, L_2) → m_3
  147. Rotate(H_1, If[30 < b ≤ 31, ((b - 30) * π), π], J_2) → M_2
  148. Rotate(K_2, If[30 < b ≤ 31, ((b - 30) * π), π], J_2) → N_2
  149. Segment(M_2, N_2) → n_3
  150. Dilate(J_2, If[31 < b ≤ 32, 32 - b, 0], D) → O_2
  151. Segment(J_2, O_2) → r_3
  152. Rotate(J_2, If[32 < b ≤ 34, ((b - 32) * π), (2 * π)], J_2) → P_2
  153. Rotate(O_2, If[32 < b ≤ 34, ((b - 32) * π), (2 * π)], J_2) → Q_2
  154. Segment(P_2, Q_2) → s_3
  155. CircleArc(J_2, D, Q_2) → t_3
  156. Circle(J_2, D) → c_4
  157. Segment(B, T) → a_3
  158. Segment(I, C_1) → b_3
  159. Segment(A, W) → f_4
  160. Segment(H_1, I) → g_4
  161. Segment(H_1, T) → h_4
  162. Segment(H_1, W) → i_4
  163. Polygon(L_1, D, E, J_1) → poly2, l_4, d_4, e_4, j_4
  164. Angle(J_1, L_1, D) → δ
  165. Angle(L_1, D, E) → ε
  166. Angle(D, E, J_1) → ζ
  167. Angle(E, J_1, L_1) → η
  168. Dilate(D, If[37 < b ≤ 38, 38 - b, 0], L_1) → D'
  169. Segment(D, D') → k_4
  170. Dilate(L_1, If[38 < b ≤ 39, 39 - b, 0], J_1) → R_2
  171. Segment(L_1, R_2) → m_4
  172. Dilate(J_1, If[39 < b ≤ 40, 40 - b, 0], E) → S_2
  173. Segment(J_1, S_2) → n_4
  174. Dilate(E, If[40 < b ≤ 41, 41 - b, 0], D) → E'
  175. Segment(E, E') → p_4
  176. Polygon(D, F, J_1, K_1) → poly3, d_5, f_5, j_5, k_5
  177. Angle(J_1, F, D) → θ
  178. Angle(F, D, K_1) → ι
  179. Angle(D, K_1, J_1) → κ
  180. Angle(K_1, J_1, F) → λ
  181. Dilate(F, If[42 < b ≤ 43, 43 - b, 0], D) → F'
  182. Segment(F, F') → q_4
  183. Dilate(D, If[43 < b ≤ 44, 44 - b, 0], K_1) → D'_1
  184. Segment(D, D'_1) → r_4
  185. Dilate(K_1, If[44 < b ≤ 45, 45 - b, 0], J_1) → T_2
  186. Segment(K_1, T_2) → s_4
  187. Dilate(J_1, If[45 < b ≤ 46, 46 - b, 0], F) → U_2
  188. Segment(J_1, U_2) → t_4
  189. Polygon(E, F, L_1, K_1) → poly4, e_5, f_6, l_5, k_6
  190. Dilate(L_1, If[47 < b ≤ 48, 48 - b, 0], F) → V_2
  191. Segment(L_1, V_2) → a_4
  192. Dilate(F, If[48 < b ≤ 49, 49 - b, 0], E) → F'_1
  193. Segment(F, F'_1) → b_4
  194. Dilate(E, If[49 < b ≤ 50, 50 - b, 0], K_1) → E'_1
  195. Segment(E, E'_1) → g_5
  196. Dilate(K_1, If[50 < b ≤ 51, 51 - b, 0], L_1) → W_2
  197. Segment(K_1, W_2) → h_5
  198. Angle(E, F, L_1) → μ
  199. Angle(F, L_1, K_1) → ν
  200. Angle(L_1, K_1, E) → ξ
  201. Angle(K_1, E, F) → ο

AI Drawing Prompt Reference

Draw a triangle, mark the midpoints of its three sides, the midpoints of the three altitudes, and the midpoints of the segments connecting each vertex to the orthocenter. Construct and draw the nine-point circle using these points, showing its relationship with the circumcircle of the triangle. Label key geometric elements like the orthocenter, midpoints, and circles, following the construction sequence.
The prompt above was compiled from the original drawing commands and is for reference only.