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Philippine Abaniko: Rotation-symmetric geometry with incircle and equilateral tr

Philippine Abaniko: Rotation-symmetric geometry with incircle and equilateral tr
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2023-09-16 04:58
TriangleCircleAngleTransformationGeometric Construction
This example demonstrates how to build a rotation-symmetric geometric figure using repeated constructions, highlighting the relationship between an incircle (inscribed circle) and equilateral-triangle structure. Key concepts include rotation and symmetry, circle and arc constructions (circumcircle arcs and circle arcs), angle bisectors, and the geometric constraints that lead to inscribed/interacting circles. Construction outline: start with Point, Intersect, and Segment to place key points and intersection nodes; apply Rotate to generate rotated copies of the fundamental parts; use Circle together with CircumcircleArc and CircleArc to draw the required circles and arcs; construct necessary angles via AngularBisector; then employ Polygon, Midpoint, and Ray to assemble triangle configurations and auxiliary elements that enforce the equilateral-triangle relationships. The repeated creation of corresponding points (e.g., A′, B′, A′1, B′1, etc.) makes the symmetry explicit and visually verifies the structure. Suitable for secondary school geometry and contest preparation, especially lessons on transformation geometry and classical constructions involving circles and angle bisectors.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Intersect(xAxis, yAxis) → B
  3. Segment(A, B) → f
  4. Rotate(A, (-α), B) → A'
  5. Rotate(B, (-α), B) → B'
  6. Segment(A', B') → f'
  7. Rotate(A, -15°, B) → A'_{1}
  8. Rotate(B, -15°, B) → B'_{1}
  9. Segment(A'_{1}, B'_{1}) → f'_{1}
  10. Rotate(A'_{1}, -15°, B) → D
  11. Rotate(B'_{1}, -15°, B) → E
  12. Segment(D, E) → g
  13. Rotate(D, -15°, B) → D'
  14. Rotate(E, -15°, B) → E'
  15. Segment(D', E') → g'
  16. Rotate(D', -15°, B) → D''
  17. Rotate(E', -15°, B) → E''
  18. Segment(D'', E'') → g''
  19. Rotate(D'', -15°, B) → D'''
  20. Rotate(E'', -15°, B) → E'''
  21. Segment(D''', E''') → g'''
  22. Rotate(D''', -15°, B) → F
  23. Rotate(E''', -15°, B) → G
  24. Segment(F, G) → h
  25. Rotate(F, -15°, B) → F'
  26. Rotate(G, -15°, B) → G'
  27. Segment(F', G') → h'
  28. Rotate(F', -15°, B) → F''
  29. Rotate(G', -15°, B) → G''
  30. Segment(F'', G'') → h''
  31. Rotate(F'', -15°, B) → F'''
  32. Rotate(G'', -15°, B) → G'''
  33. Segment(F''', G''') → h'''
  34. Rotate(F''', -15°, B) → H
  35. Rotate(G''', -15°, B) → I
  36. Segment(H, I) → i
  37. Rotate(H, -15°, B) → H'
  38. Rotate(I, -15°, B) → I'
  39. Segment(H', I') → i'
  40. Point(f) → J
  41. Circle(B, J) → c
  42. Intersect(c, f', 1) → K
  43. Intersect(c, f'_{1}, 1) → L
  44. CircumcircleArc(J, L, K) → d
  45. AngularBisector(A'_{1}, B, A) → j
  46. AngularBisector(D, B, A'_{1}) → k
  47. Circle(B, A'_{1}) → e
  48. Polygon(A'_{1}, A, J, L) → q1, a'_{1}, a, j_1, l_1
  49. Rotate(A'_{1}, -15°, B) → P
  50. Rotate(A, -15°, B) → A'_{3}
  51. Rotate(J, -15°, B) → J'
  52. Rotate(L, -15°, B) → L'
  53. Polygon(P, A'_{3}, J', L') → q1', p, a'_{3}, j', l'
  54. Rotate(P, -15°, B) → P'
  55. Rotate(A'_{3}, -15°, B) → Q
  56. Rotate(J', -15°, B) → J''
  57. Rotate(L', -15°, B) → L''
  58. Polygon(P', Q, J'', L'') → q1'', p', q, j'', l''
  59. Rotate(P', -15°, B) → P''
  60. Rotate(Q, -15°, B) → Q'
  61. Rotate(J'', -15°, B) → J'''
  62. Rotate(L'', -15°, B) → L'''
  63. Polygon(P'', Q', J''', L''') → q1''', p'', q', j''', l'''
  64. Rotate(P'', -15°, B) → P'''
  65. Rotate(Q', -15°, B) → Q''
  66. Rotate(J''', -15°, B) → R
  67. Rotate(L''', -15°, B) → S
  68. Polygon(P''', Q'', R, S) → q2, p''', q'', r, s
  69. Rotate(P''', -15°, B) → T
  70. Rotate(Q'', -15°, B) → Q'''
  71. Rotate(R, -15°, B) → R'
  72. Rotate(S, -15°, B) → S'
  73. Polygon(T, Q''', R', S') → q2', t, q''', r', s'
  74. Rotate(T, -15°, B) → T'
  75. Rotate(Q''', -15°, B) → U
  76. Rotate(R', -15°, B) → R''
  77. Rotate(S', -15°, B) → S''
  78. Polygon(T', U, R'', S'') → q2'', t', u, r'', s''
  79. Rotate(T', -15°, B) → T''
  80. Rotate(U, -15°, B) → U'
  81. Rotate(R'', -15°, B) → R'''
  82. Rotate(S'', -15°, B) → S'''
  83. Polygon(T'', U', R''', S''') → q2''', t'', u', r''', s'''
  84. Rotate(T'', -15°, B) → T'''
  85. Rotate(U', -15°, B) → U''
  86. Rotate(R''', -15°, B) → V
  87. Rotate(S''', -15°, B) → W
  88. Polygon(T''', U'', V, W) → q3, t''', u'', v, w
  89. Rotate(T''', -15°, B) → Z
  90. Rotate(U'', -15°, B) → U'''
  91. Rotate(V, -15°, B) → V'
  92. Rotate(W, -15°, B) → W'
  93. Polygon(Z, U''', V', W') → q3', z_1, u''', v', w'
  94. Rotate(Z, -15°, B) → Z'
  95. Rotate(U''', -15°, B) → A_1
  96. Rotate(V', -15°, B) → V''
  97. Rotate(W', -15°, B) → W''
  98. Polygon(Z', A_1, V'', W'') → q3'', z', a_1, v'', w''
  99. Rotate(Z', -15°, B) → Z''
  100. Rotate(A_1, -15°, B) → B_1
  101. Rotate(V'', -15°, B) → V'''
  102. Rotate(W'', -15°, B) → W'''
  103. Polygon(W'', K, A', H') → q4, n, k_1, a', b
  104. Polygon(A', A, J, K) → q5, l, a_2, j_2, k_2
  105. Polygon(A', D, L', K) → q6, m, d_1, f_1, k_3
  106. Polygon(A', A'_{1}, L, K) → q7, g_1, h_1, l_2, k_4
  107. Polygon(D', L'', K, A') → q8, d', i_1, k_5, m_1
  108. Polygon(D'', A', K, L''') → q9, d'', n_1, k_6, p_1
  109. Polygon(D''', S, K, A') → q10, d''', s_1, k_7, q_1
  110. Polygon(F, A', K, S') → q11, f_2, r_1, k_8, t_1
  111. Polygon(F', S'', K, A') → q12, b_1, c_1, k_9, e_1
  112. Polygon(F'', A', K, S''') → q13, f'', g_2, k_{10}, h_2
  113. Polygon(F''', A', K, W) → q14, f''', i_2, k_{11}, w_1
  114. Polygon(W', K, A', H) → q15, m_2, k_{12}, n_2, h_3
  115. Midpoint(D, A'_{1}) → M
  116. Point(f) → N
  117. Circle(B, N) → c_2
  118. Intersect(c_2, f', 1) → O
  119. Intersect(c_2, f'_{1}, 1) → C_1
  120. CircumcircleArc(N, C_1, O) → d_2
  121. CircleArc(B, O, N) → e_2
  122. Point(f) → D_1
  123. Circle(B, D_1) → p_2
  124. Intersect(p_2, f'_{1}, 1) → E_1
  125. Intersect(p_2, g, 1) → F_1
  126. Intersect(p_2, g', 1) → G_1
  127. Intersect(p_2, g'', 1) → H_1
  128. Intersect(p_2, g''', 1) → I_1
  129. Intersect(p_2, h, 1) → J_1
  130. Intersect(p_2, h', 1) → K_1
  131. Intersect(p_2, h'', 1) → L_1
  132. Intersect(p_2, h''', 1) → M_1
  133. Intersect(p_2, i, 1) → N_1
  134. Intersect(p_2, i', 1) → O_1
  135. Intersect(p_2, f', 1) → P_1
  136. Ray(P_1, B) → q_2
  137. Ray(E_1, B) → r_2
  138. Ray(F_1, B) → s_2
  139. Ray(G_1, B) → t_2
  140. Ray(H_1, B) → b_2
  141. Ray(I_1, B) → f_3
  142. Ray(J_1, B) → g_3
  143. Ray(K_1, B) → i_3
  144. Ray(L_1, B) → j_3
  145. Ray(M_1, B) → l_3
  146. Ray(N_1, B) → m_3
  147. Ray(O_1, B) → n_3
  148. Intersect(p_2, n_3, 2) → Q_1
  149. Intersect(p_2, m_3, 2) → R_1
  150. Intersect(p_2, l_3, 2) → S_1
  151. Intersect(p_2, j_3, 2) → T_1
  152. Intersect(p_2, i_3, 2) → U_1
  153. Intersect(p_2, g_3, 2) → V_1
  154. Intersect(p_2, f_3, 2) → W_1
  155. Intersect(p_2, b_2, 2) → Z_1
  156. Intersect(p_2, t_2, 2) → A_2
  157. Intersect(p_2, s_2, 2) → B_2
  158. Intersect(p_2, r_2, 2) → C_2
  159. Segment(E_1, C_2) → p_3
  160. Ray(P_1, B) → q_3
  161. Intersect(p_2, q_3, 2) → D_2
  162. Segment(P_1, D_2) → r_3
  163. Segment(F_1, B_2) → s_3
  164. Segment(A_2, G_1) → t_3
  165. Segment(Z_1, H_1) → a_3
  166. Segment(W_1, I_1) → b_3
  167. Segment(V_1, J_1) → c_3
  168. Segment(U_1, K_1) → d_3
  169. Segment(T_1, L_1) → e_3
  170. Segment(S_1, M_1) → f_4
  171. Segment(R_1, N_1) → g_4
  172. Segment(Q_1, O_1) → h_4

AI Drawing Prompt Reference

Draw a rotation-symmetric geometric figure with an incircle and equilateral triangles. Start by placing point A on the x-axis and point B at the intersection of the x-axis and y-axis, then connect A and B. Rotate A around B by -15° to get A' and B', and connect A' and B'. Repeat this process, rotating -15° each time, to generate a series of equilateral triangles and incircles, forming a rotation-symmetric figure.
The prompt above was compiled from the original drawing commands and is for reference only.