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9-Point Circle

9-Point Circle
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2016-05-29 11:51
TriangleCircleGeometric Construction
This example demonstrates the 9-Point Circle of a Triangle. The 9-Point Circle is a circle that passes through nine significant points of a triangle: the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the three segments joining the vertices to the orthocenter. Key mathematical concepts include midpoints, feet of altitudes, midlines, and circles. Using GeoGebra commands such as Segment, Midpoint, Circle, etc., the construction of the 9-Point Circle is illustrated. Suitable for students at the junior high school level and above for geometric construction.

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, (0.15 * a)) → c
  9. Circle(E, (0.15 * a)) → d
  10. Circle(F, (0.15 * 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)) * pi), (-pi)], D) → A'_1
  21. Rotate(A', If[1 < b ≤ 2, ((b - 1) * pi), pi], D) → A''
  22. Segment(A'_1, A'') → l'
  23. Rotate(B, If[3 < b ≤ 4, ((-(b - 3)) * pi), (-pi)], F) → B'_1
  24. Rotate(B', If[3 < b ≤ 4, ((b - 3) * pi), pi], F) → B''
  25. Segment(B'_1, B'') → m'
  26. Rotate(C_1, If[5 < b ≤ 6, ((-(b - 5)) * pi), (-pi)], E) → C'_1
  27. Rotate(C', If[5 < b ≤ 6, ((b - 5) * pi), pi], 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, (0.15 * a)) → s
  42. Intersect(s, i, 2) → N
  43. Intersect(s, j, 2) → O
  44. Angle(C_1, B, A) → α
  45. Angle(B, A, C_1) → β
  46. Angle(A, C_1, B) → γ
  47. Line(A, C_1) → f_1
  48. Point(f_1) → P
  49. Point(f_1) → Q
  50. Ray(A, P) → g_1
  51. Ray(C_1, Q) → h_1
  52. Point(g_1) → R
  53. Ray(A, R) → i_1
  54. Point(h_1) → S
  55. Ray(C_1, S) → j_1
  56. OrthogonalLine(B, f_1) → k_1
  57. Intersect(f_1, k_1) → T
  58. Circle(T, (0.15 * a)) → d_1
  59. Intersect(d_1, f_1, 1) → U
  60. Intersect(d_1, k_1, 1) → V
  61. Line(C_1, B) → l_1
  62. OrthogonalLine(A, l_1) → m_1
  63. Intersect(l_1, m_1) → W
  64. Circle(W, (0.15 * a)) → p_1
  65. Intersect(p_1, l_1, 1) → Z
  66. Intersect(p_1, m_1, 1) → A_1
  67. Point(l_1) → B_1
  68. Point(l_1) → D_1
  69. Ray(C_1, B_1) → n_1
  70. Ray(B, D_1) → r_1
  71. Point(n_1) → E_1
  72. Point(r_1) → F_1
  73. Ray(C_1, E_1) → s_1
  74. Ray(B, F_1) → t_1
  75. Dilate(C_1, If[6 < b ≤ 7, 7 - b, 0], I) → G_1
  76. Segment(C_1, 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) → H_1
  82. Dilate(I, If[12 < b ≤ 13, 13 - b, 0], H_1) → I'
  83. Dilate(A, If[12 < b ≤ 13, 13 - b, 0], H_1) → A'_3
  84. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], H_1) → T'
  85. Segment(I, I') → i_2
  86. Segment(A, A'_3) → j_2
  87. Segment(T, T') → k_2
  88. Dilate(C_1, If[12 < b ≤ 13, 13 - b, 0], H_1) → I_1
  89. Dilate(W, If[12 < b ≤ 13, 13 - b, 0], H_1) → W'
  90. Dilate(T, If[12 < b ≤ 13, 13 - b, 0], H_1) → T'_1
  91. Segment(W, W') → l_2
  92. Segment(C_1, I_1) → m_2
  93. Segment(T, T'_1) → n_2
  94. Dilate(I, If[12 < b ≤ 13, 13 - b, 0], H_1) → I'_1
  95. Dilate(B, If[12 < b ≤ 13, 13 - b, 0], H_1) → B'_3
  96. Dilate(W, If[12 < b ≤ 13, 13 - b, 0], H_1) → 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(H_1, C_1) → J_1
  101. Midpoint(H_1, A) → K_1
  102. Midpoint(H_1, B) → L_1
  103. Dilate(H_1, If[14 < b ≤ 15, 15 - b, 0], J_1) → M_1
  104. Segment(H_1, M_1) → s_2
  105. Dilate(J_1, If[15 < b ≤ 16, 16 - b, 0], C_1) → N_1
  106. Segment(J_1, N_1) → t_2
  107. Rotate(H_1, If[16 < b ≤ 17, ((b - 16) * pi), pi], J_1) → O_1
  108. Rotate(M_1, If[16 < b ≤ 17, ((b - 16) * pi), pi], J_1) → P_1
  109. Segment(O_1, P_1) → a_2
  110. Dilate(H_1, If[17 < b ≤ 18, 18 - b, 0], L_1) → Q_1
  111. Segment(H_1, Q_1) → b_2
  112. Dilate(L_1, If[18 < b ≤ 19, 19 - b, 0], B) → R_1
  113. Segment(L_1, R_1) → e_2
  114. Rotate(H_1, If[19 < b ≤ 20, ((b - 19) * pi), pi], L_1) → S_1
  115. Rotate(Q_1, If[19 < b ≤ 20, ((b - 19) * pi), pi], L_1) → T_1
  116. Segment(S_1, T_1) → f_3
  117. Dilate(H_1, If[20 < b ≤ 21, 21 - b, 0], K_1) → U_1
  118. Segment(H_1, U_1) → g_3
  119. Dilate(K_1, If[21 < b ≤ 22, 22 - b, 0], A) → V_1
  120. Segment(K_1, V_1) → h_3
  121. Rotate(H_1, If[22 < b ≤ 23, ((b - 22) * pi), pi], K_1) → W_1
  122. Rotate(U_1, If[22 < b ≤ 23, ((b - 22) * pi), pi], K_1) → Z_1
  123. Segment(W_1, Z_1) → i_3
  124. Intersect(e, l_1) → A_2, B_2
  125. OrthogonalLine(F, l_1) → j_3
  126. Intersect(e, j_3) → C_2, D_2
  127. OrthogonalLine(E, f_1) → k_3
  128. Circle(E, (0.15 * a)) → d_3
  129. Intersect(d_3, k_3) → E_2, F_2
  130. Intersect(d_3, f_1) → G_2, H_2
  131. Intersect(k_3, j_3) → I_2
  132. Midpoint(H_1, I_2) → J_2
  133. Dilate(H_1, If[28 < b ≤ 29, 29 - b, 0], J_2) → K_2
  134. Segment(H_1, K_2) → l_3
  135. Dilate(J_2, If[29 < b ≤ 30, 30 - b, 0], I_2) → L_2
  136. Segment(J_2, L_2) → m_3
  137. Rotate(H_1, If[30 < b ≤ 31, ((b - 30) * pi), pi], J_2) → M_2
  138. Rotate(K_2, If[30 < b ≤ 31, ((b - 30) * pi), pi], J_2) → N_2
  139. Segment(M_2, N_2) → n_3
  140. Dilate(J_2, If[31 < b ≤ 32, 32 - b, 0], D) → O_2
  141. Segment(J_2, O_2) → r_3
  142. Rotate(J_2, If[32 < b ≤ 34, ((b - 32) * pi), (2 * pi)], J_2) → P_2
  143. Rotate(O_2, If[32 < b ≤ 34, ((b - 32) * pi), (2 * pi)], J_2) → Q_2
  144. Segment(P_2, Q_2) → s_3
  145. CircleArc(J_2, D, Q_2) → t_3
  146. Circle(J_2, D) → c_4
  147. Segment(B, T) → a_3
  148. Segment(I, C_1) → b_3
  149. Segment(A, W) → f_4
  150. Segment(H_1, I) → g_4
  151. Segment(H_1, T) → h_4
  152. Segment(H_1, W) → i_4
  153. Angle(B, T, C_1) → δ
  154. Angle(B, I, C_1) → ε
  155. Angle(A, W, B) → ζ
  156. Angle(I_2, E, C_1) → η
  157. Angle(I_2, F, B) → θ