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9-Point Circle Action (Part 2): Circumradius is Double the Nine-Point Radius

9-Point Circle Action (Part 2): Circumradius is Double the Nine-Point Radius
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2016-05-31 10:30
TriangleCircleTransformationGeometric ConstructionLocus
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 dynamic geometry example visually demonstrates a fundamental theorem: the radius of a triangle's circumcircle is exactly twice the radius of its nine-point circle. The nine-point circle passes through the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments from the orthocenter to each vertex. The construction uses GeoGebra commands including Line, Segment, Midpoint, OrthogonalLine, Intersect, Circle, Dilate, Rotate, Polygon, Point, Ray, CircleSector, Angle, CircleArc, Vector, and Translate to build the triangle, its circumcircle, the nine-point circle, and related points. This example is ideal for high school and college students and teachers exploring triangle geometry, circle properties, and visual proofs without words. It helps learners grasp the radius ratio and the relationship between the circumcenter, orthocenter, and nine-point center.

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. Segment(I_2, C_1) → d_2
  155. Dilate(J_2, If[1 < v ≤ 2, 2 - v, 0], O) → H_1
  156. Segment(J_2, H_1) → f_3
  157. Rotate(I_2, If[2 < v ≤ 3, ((v - 2) * π), π], J_2) → K_1
  158. Rotate(C_1, If[2 < v ≤ 3, ((v - 2) * π), π], J_2) → V_2
  159. Segment(K_1, V_2) → g_3
  160. Dilate(J_2, If[3 < v ≤ 4, 4 - v, 0], D_3) → W_2
  161. Segment(J_2, W_2) → i_3
  162. Rotate(J_2, If[4 < v ≤ 6, ((v - 4) * π), (2 * π)], J_2) → Z_2
  163. Rotate(W_2, If[4 < v ≤ 6, ((v - 4) * π), (2 * π)], J_2) → A_3
  164. Segment(Z_2, A_3) → l_3
  165. CircleArc(J_2, D_3, A_3) → p_3
  166. Dilate(I_2, If[6 < v ≤ 7, 7 - v, 0], I_3) → L_1
  167. Dilate(I_3, If[7 < v ≤ 8, 8 - v, 0], C) → B_3
  168. Segment(I_2, L_1) → m_3
  169. Segment(I_3, B_3) → n_3
  170. Dilate(J_2, If[8 < v ≤ 9, 9 - v, 0], I_2) → G_3
  171. Vector(J_2, G_3) → w
  172. Translate(Z_2, w) → H_3
  173. Translate(A_3, w) → J_3
  174. Segment(H_3, J_3) → q_3
  175. Rotate(H_3, If[9 < v ≤ 10, ((v - 9) * π), π], I_2) → K_3
  176. Rotate(J_3, If[9 < v ≤ 10, ((v - 9) * π), π], I_2) → L_3
  177. Segment(K_3, L_3) → r_3
  178. Rotate(K_3, If[10 < v ≤ 11, ((v - 10) * π), π], I_3) → M_3
  179. Rotate(L_3, If[10 < v ≤ 11, ((v - 10) * π), π], I_3) → N_3
  180. Segment(M_3, N_3) → g_4
  181. Midpoint(I_2, A) → R_3
  182. Midpoint(I_2, B) → S_3
  183. Dilate(I_2, If[6 < v ≤ 7, 7 - v, 0], R_3) → U_3
  184. Dilate(I_2, If[6 < v ≤ 7, 7 - v, 0], S_3) → W_3
  185. Segment(I_2, U_3) → a_3
  186. Segment(I_2, W_3) → b_3
  187. Dilate(R_3, If[7 < v ≤ 8, 8 - v, 0], A) → J_1
  188. Dilate(S_3, If[7 < v ≤ 8, 8 - v, 0], B) → B_4
  189. Segment(R_3, J_1) → f_4
  190. Segment(S_3, B_4) → h_4
  191. Line(J_2, a_3) → i_4
  192. Dilate(J_2, If[11 < v ≤ 12, 12 - v, 0], I_2) → D_4
  193. Vector(J_2, D_4) → u_1
  194. Translate(Z_2, u_1) → G_4
  195. Translate(A_3, u_1) → H_4
  196. Segment(G_4, H_4) → l_4
  197. Angle(D, J_2, D_3) → μ
  198. Rotate(G_4, If[12 < v ≤ 13, ((-(v - 12)) * μ), (-μ)], I_2) → P_3
  199. Rotate(H_4, If[12 < v ≤ 13, ((-(v - 12)) * μ), (-μ)], I_2) → Q_3
  200. Segment(P_3, Q_3) → m_4
  201. Rotate(P_3, If[13 < v ≤ 14, ((-(v - 13)) * π), (-π)], R_3) → T_3
  202. Rotate(Q_3, If[13 < v ≤ 14, ((-(v - 13)) * π), (-π)], R_3) → V_3
  203. Segment(T_3, V_3) → n_4
  204. Dilate(J_2, If[14 < v ≤ 15, 15 - v, 0], I_2) → Z_3
  205. Vector(J_2, Z_3) → v_1
  206. Translate(Z_2, v_1) → A_4
  207. Translate(A_3, v_1) → C_4
  208. Segment(A_4, C_4) → p_4
  209. Line(J_2, b_3) → q_4
  210. Angle(D_3, J_2, E) → κ
  211. Rotate(A_4, If[15 < v ≤ 16, ((v - 15) * κ), κ], I_2) → E_4
  212. Rotate(C_4, If[15 < v ≤ 16, ((v - 15) * κ), κ], I_2) → F_4
  213. Segment(E_4, F_4) → r_4
  214. Rotate(E_4, If[16 < v ≤ 17, ((v - 16) * π), π], S_3) → I_4
  215. Rotate(F_4, If[16 < v ≤ 17, ((v - 16) * π), π], S_3) → J_4
  216. Segment(I_4, J_4) → s_4
  217. Circle(I_2, C) → d_4
  218. Rotate(B, If[17 < v ≤ 19, ((v - 17) * π), (2 * π)], I_2) → B'_4
  219. CircleArc(I_2, B, B'_4) → e_4
  220. Line(B, C) → q
  221. Line(A, C) → r
  222. Line(B, A) → i_1