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9-Point Circle Property: Midpoint of Line Segment from Orthocenter to Circumcirc

9-Point Circle Property: Midpoint of Line Segment from Orthocenter to Circumcirc
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2016-05-31 11:07
TriangleCircleGeometric 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 Triangle Theorems Triangles: Points of Concurrency
This example demonstrates a key property of the 9-Point Circle: the midpoint of the line segment connecting the orthocenter of a triangle to any point on its circumcircle lies on the 9-Point Circle. Key concepts include the 9-Point Circle, orthocenter, and midpoint. GeoGebra commands used include Line, Segment, Midpoint, Circle, Dilate, Rotate, Polygon, Point, etc. Suitable for high school geometry education.

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
  166. Translate(Z_2, w) → H_3
  167. Translate(A_3, w) → 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(d_4) → K_4

AI Drawing Prompt Reference

Draw a triangle ABC. Construct line f1 from A to C and line segment h from C to A, with midpoint E3. Construct line f from B to C and line segment h, with midpoint F3. Construct a perpendicular line j3 from E3 to f1 and a perpendicular line k3 from F3 to l1, and let their intersection I2 be a point on the 9-point circle. Construct line segment f from A to B, with midpoint D3. Draw a circle c centered at D3 with radius 0.15 times the length of AB. Repeat the process to draw circles d and e centered at E3 and F3, respectively, with the same radius. Let the intersections of c with f, d with h, and e with g be points H, K, and M, respectively, on the 9-point circle.
The prompt above was compiled from the original drawing commands and is for reference only.