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Clustered Circles with Single Intersection Points (Veu110)

Clustered Circles with Single Intersection Points (Veu110)
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2019-11-16 05:36
CircleAngleTransformationGeometric ConstructionLocus
① This example demonstrates a cluster of circles that intersect each other at exactly one point, creating a dense yet orderly geometric pattern. ② The core mathematical concepts involve circle intersection properties, rotation and scaling transformations, and precise geometric relations using midpoints, distances, and angles. ③ Construction highlights: using GeoGebra commands such as Circle, Rotate, Dilate, Midpoint, Intersect, Segment, Angle, Line, and Vector to generate multiple rotated and scaled circles, with auxiliary elements controlling the intersection points to form the clustered arrangement. ④ Suitable for high school or college geometry classes, useful for exploring circle intersection patterns, transformation combinations, and parametric design in dynamic geometry.

GeoGebra Commands & Parameters

  1. Circle(A, 2.613126) → c
  2. Rotate(B, 45°, A) → B'
  3. Rotate(B', 45°, A) → B''
  4. Rotate(B'', 45°, A) → B'''
  5. Rotate(B''', 45°, A) → C
  6. Rotate(C, 45°, A) → C'
  7. Rotate(C', 45°, A) → C''
  8. Rotate(C'', 45°, A) → C'''
  9. Circle(B', 1) → d
  10. Circle(B'', 1) → e
  11. Circle(B''', 1) → f
  12. Circle(C, 1) → g
  13. Circle(C', 1) → h
  14. Circle(C'', 1) → k
  15. Circle(C''', 1) → p
  16. Circle(B, 1) → q
  17. Circle(A, 1.613126) → r
  18. Segment(B, B') → i
  19. Segment(B', B'') → j
  20. Segment(B'', B''') → l
  21. Segment(B''', C) → m
  22. Segment(C, C') → n
  23. Segment(C', C'') → s
  24. Segment(C'', C''') → t
  25. Segment(C''', B) → a
  26. Segment(A, B') → b
  27. Segment(A, B) → f_1
  28. Segment(A, B'') → g_1
  29. Segment(A, B''') → h_1
  30. Segment(A, C) → i_1
  31. Segment(A, C') → j_1
  32. Segment(A, C'') → k_1
  33. Segment(C''', A) → l_1
  34. Angle(B, A, B') → α
  35. Midpoint(i) → D
  36. Segment(A, E) → m_1
  37. Dilate(A, 1.717438935, A) → A'
  38. Dilate(E, 1.717438935, A) → E'
  39. Segment(A', E') → n_1
  40. Circle(E', 1) → c_1
  41. Intersect(c_1, d) → F, G, ,
  42. Rotate(c_1, 45°, A) → d_1
  43. Rotate(d_1, 45°, A) → e_1
  44. Rotate(e_1, 45°, A) → p_1
  45. Rotate(p_1, 45°, A) → q_1
  46. Rotate(q_1, 45°, A) → r_1
  47. Rotate(r_1, 45°, A) → s_1
  48. Rotate(s_1, 45°, A) → t_1
  49. Segment(B, B'') → a_1
  50. Midpoint(a_1) → H
  51. Distance(H, B) → afstandHB
  52. Segment(H, B) → b_1
  53. Rotate(H, 90°, H) → H'
  54. Rotate(B, 90°, H) → B'_1
  55. Segment(H', B'_1) → f_2
  56. Dilate(H', 1.732, H) → H''
  57. Dilate(B'_1, 1.732, H) → I
  58. Segment(H'', I) → g_2
  59. Circle(I, 1) → c_2
  60. Rotate(c_2, 45°, A) → d_2
  61. Rotate(d_2, 45°, A) → e_2
  62. Rotate(e_2, 45°, A) → h_2
  63. Rotate(h_2, 45°, A) → k_2
  64. Rotate(k_2, 45°, A) → p_2
  65. Rotate(p_2, 45°, A) → q_2
  66. Rotate(q_2, 45°, A) → r_2
  67. Intersect(c_2, c_1) → N, O, ,
  68. Circle(A, K) → s_2
  69. Circle(A, 4.132155682847749) → t_2
  70. Circle(A, L) → c_3
  71. Circle(P, M) → d_3
  72. Intersect(d_1, d_3) → S, T, ,
  73. Rotate(d_3, 45°, A) → e_3
  74. Rotate(e_3, 45°, A) → f_3
  75. Rotate(f_3, 45°, A) → g_3
  76. Rotate(g_3, 45°, A) → h_3
  77. Rotate(h_3, 45°, A) → k_3
  78. Rotate(k_3, 45°, A) → p_3
  79. Rotate(p_3, 45°, A) → q_3
  80. Intersect(e_3, d_1) → U, V, ,
  81. Intersect(e_1, c_3, 1) → W
  82. Intersect(e_3, e_1) → Z, A_1, ,
  83. Intersect(f_3, e_1) → B_1, C_1, ,
  84. Intersect(f_3, p_1) → D_1, E_1, ,
  85. Intersect(d_1, d) → F_1, G_1, ,
  86. Intersect(e, d_1) → H_1, I_1, ,
  87. Intersect(e_1, e) → J_1, K_1, ,
  88. Intersect(e_1, f) → L_1, M_1, ,
  89. Intersect(f, p_1) → N_1, O_1, ,
  90. Intersect(p_1, g_3) → P_1, Q_1, ,
  91. Intersect(g_3, q_1) → R_1, S_1, ,
  92. Intersect(g, f) → T_1, , ,
  93. Intersect(g, p_1) → U_1, V_1, ,
  94. Intersect(q_1, h) → W_1, Z_1, ,
  95. Intersect(q_1, h_3) → A_2, B_2, ,
  96. Intersect(h_3, r_1) → C_2, D_2, ,
  97. Intersect(h, r_1) → E_2, F_2, ,
  98. Intersect(k_3, r_1) → G_2, H_2, ,
  99. Intersect(s_1, k_3) → I_2, J_2, ,
  100. Intersect(p_3, s_1) → K_2, L_2, ,
  101. Intersect(p_3, t_1) → M_2, N_2, ,
  102. Intersect(q_3, c_1) → O_2, P_2, ,
  103. Intersect(q_3, t_1) → Q_2, R_2, ,
  104. Intersect(p, t_1) → S_2, T_2, ,
  105. Intersect(s_1, p) → U_2, V_2, ,
  106. Intersect(k, s_1) → W_2, Z_2, ,
  107. Intersect(e_3, s_2) → A_3, B_3, ,
  108. Intersect(f_3, s_2) → C_3, D_3, ,
  109. Intersect(d_3, s_2) → E_3, F_3, ,
  110. Intersect(g_3, s_2) → G_3, H_3, ,
  111. Intersect(h_3, s_2) → I_3, J_3, ,
  112. Intersect(k_3, s_2) → K_3, L_3, ,
  113. Intersect(p_3, s_2) → M_3, N_3, ,
  114. Intersect(q_3, s_2) → O_3, P_3, ,
  115. Line(C', B') → i_2
  116. Line(C''', B''') → j_2
  117. Line(B, C) → l_2
  118. Line(B'', C'') → m_2
  119. Intersect(k, r_1) → Q_3, R_3, ,
  120. Intersect(g, q_1) → S_3, T_3, ,
  121. Intersect(q, c_1) → U_3, V_3, ,
  122. Intersect(t_1, q) → W_3, Z_3, ,
  123. Circle(A_4, 1) → r_3
  124. Circle(P, 2.0144333173267204) → s_3
  125. Intersect(d, r_3) → B_4, C_4, ,
  126. Intersect(r, j_2) → D_4, E_4
  127. Intersect(r, i_2) → F_4, G_4
  128. Intersect(r, l_2) → H_4, I_4
  129. Intersect(r, m_2) → J_4, K_4
  130. Circle(A, 0.5) → t_3
  131. Circle(L_4, 1.8849655) → c_4
  132. Intersect(c_4, q_3) → M_4, H_5, ,
  133. Intersect(c_4, d_3) → N_4, I_5, ,
  134. Intersect(c_4, c_3) → O_4, , ,
  135. Rotate(c_4, 45°, A) → d_4
  136. Rotate(d_4, 45°, A) → e_4
  137. Rotate(e_4, 45°, A) → f_4
  138. Rotate(f_4, 45°, A) → g_4
  139. Rotate(g_4, 45°, A) → h_4
  140. Rotate(h_4, 45°, A) → k_4
  141. Rotate(k_4, 45°, A) → p_4
  142. Intersect(d_4, d_3) → P_4, J_5, ,
  143. Intersect(d_4, e_3) → Q_4, K_5, ,
  144. Intersect(e_4, e_3) → R_4, L_5, ,
  145. Intersect(e_4, f_3) → S_4, M_5, ,
  146. Intersect(e_4, c_3) → T_4, , ,
  147. Intersect(c_3, d_4) → U_4, , ,
  148. Intersect(f_4, f_3) → V_4, N_5, ,
  149. Intersect(f_4, g_3) → W_4, O_5, ,
  150. Intersect(g_4, g_3) → Z_4, P_5, ,
  151. Intersect(g_4, h_3) → A_5, Q_5, ,
  152. Intersect(h_4, h_3) → B_5, R_5, ,
  153. Intersect(k_3, h_4) → C_5, S_5, ,
  154. Intersect(k_4, k_3) → D_5, T_5, ,
  155. Intersect(p_4, p_3) → E_5, U_5, ,
  156. Intersect(k_4, p_3) → F_5, V_5, ,
  157. Intersect(p_4, q_3) → G_5, W_5, ,
  158. Intersect(f_4, c_3) → Z_5, , ,
  159. Intersect(g_4, c_3) → A_6, , ,
  160. Intersect(h_4, c_3) → B_6, , ,
  161. Intersect(k_4, c_3) → C_6, , ,
  162. Intersect(p_4, c_3) → D_6, , ,
  163. Intersect(d, e) → E_6, , ,
  164. Intersect(e, f) → F_6, , ,
  165. Intersect(q, p) → G_6, , ,
  166. Intersect(k, h) → H_6, , ,
  167. Intersect(q, d) → I_6, , ,
  168. Intersect(k, p) → J_6, , ,
  169. Intersect(g, h) → K_6, , ,
  170. Rotate(f, 45°, B''') → f'
  171. Intersect(f, f', 1) → L_6
  172. Rotate(f', 45°, A) → f''
  173. Intersect(f, f', 1) → M_6
  174. Intersect(f, f', 1) → N_6
  175. Intersect(f, f', 1) → O_6
  176. Intersect(f'', f) → P_6, , ,
  177. Circle(S, U, Z) → q_4
  178. Circle(N_1, J_1, F_1) → r_4
  179. Circle(P_6, E_6, I_6) → s_4
  180. Circle(S_4, R_4, Q_4) → t_4
  181. Circle(R, 1.634479683) → d_5
  182. Intersect(c_4, d_5) → Q_6, R_6, ,
  183. Intersect(d_4, d_5) → S_6, T_6, ,
  184. Circle(A, L_4) → c_5
  185. Intersect(d_5, c_5) → U_6, V_6, ,
  186. Rotate(d_5, 45°, A) → e_5
  187. Rotate(e_5, 45°, A) → f_5
  188. Intersect(e_5, d_4) → W_6, Z_6, ,
  189. Intersect(e_4, e_5) → A_7, B_7, ,
  190. Intersect(e_5, c_5) → C_7, D_7, ,
  191. Intersect(f_5, c_5) → G_7, H_7, ,
  192. Intersect(f_5, e_4) → E_7, F_7, ,
  193. Rotate(f_5, 45°, A) → g_5
  194. Intersect(f_5, f_4) → I_7, J_7, ,
  195. Intersect(f_4, g_5) → K_7, L_7, ,
  196. Rotate(g_5, 45°, A) → h_5
  197. Rotate(h_5, 45°, A) → k_5
  198. Intersect(g_5, g_4) → M_7, N_7, ,
  199. Intersect(g_4, h_5) → O_7, P_7, ,
  200. Intersect(h_5, h_4) → Q_7, R_7, ,
  201. Intersect(h_4, k_5) → S_7, T_7, ,
  202. Intersect(k_5, k_4) → U_7, V_7, ,
  203. Rotate(k_5, 45°, A) → p_5
  204. Rotate(p_5, 45°, A) → q_5
  205. Intersect(k_5, p_5) → W_7, , ,
  206. Intersect(k_4, p_5) → Z_7, A_8, ,
  207. Intersect(p_4, p_5) → B_8, C_8, ,
  208. Intersect(p_4, q_5) → D_8, E_8, ,
  209. Intersect(q_5, c_4) → F_8, G_8, ,
  210. Intersect(c_5, g_5) → H_8, I_8, ,
  211. Intersect(c_5, h_5) → J_8, K_8, ,
  212. Intersect(c_5, p_5) → L_8, M_8, ,
  213. Intersect(c_5, k_5) → N_8, O_8, ,
  214. Intersect(q_5, c_5) → P_8, Q_8, ,
  215. Intersect(r_3, d_3) → R_8, S_8, ,
  216. Circle(F_6, E_6, I_6) → r_5
  217. Circle(A, R) → s_5
  218. Line(C_7, O_8) → n_2
  219. Line(H_8, Q_8) → a_2
  220. Circle(A, 10.48172643) → t_5
  221. Rotate(R, 45°, A) → R'
  222. Rotate(R', 45°, A) → R''
  223. Rotate(R'', 45°, A) → R'''
  224. Rotate(R''', 45°, A) → T_8
  225. Rotate(T_8, 45°, A) → U_8
  226. Rotate(U_8, 45°, A) → V_8
  227. Rotate(V_8, 45°, A) → W_8
  228. Vector(T_8, K_8) → u
  229. Dilate(u, 1.3, T_8) → u'
  230. Vector(R, U_6) → v
  231. Dilate(v, 1.3, R) → v'
  232. Vector(V_8, L_8) → w
  233. Dilate(w, 1.34, V_8) → w'
  234. Vector(R''', H_8) → u_1
  235. Vector(R'', G_7) → v_1
  236. Dilate(v_1, 1.3, R'') → w_1