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Reflections over a Circle

Reflections over a Circle
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2018-04-29 11:54
CirclePolygonTransformation
In collections Reflections over a circle
This example demonstrates reflections over a circle. Key mathematical concepts include reflection, piecewise functions, and quadrilaterals. Using GeoGebra commands such as Circle, Line, OrthogonalLine, Rotate, Tangent, Intersect, Polygon, and Mirror, complex geometric shapes can be constructed. Suitable for high school mathematics education and learning about reflection transformations.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Line(A, B) → f
  3. OrthogonalLine(A, f) → g
  4. Rotate(f, 45°, A) → f'
  5. Rotate(g, 45°, A) → g'
  6. Rotate(f, 22.5°, A) → f'_1
  7. Rotate(f'_1, 45°, A) → h
  8. Rotate(h, 45°, A) → h'
  9. Rotate(h', 45°, A) → h''
  10. Tangent(g', c) → i, j
  11. Tangent(h', c) → k, l
  12. Tangent(h'', c) → m, n
  13. Tangent(h, c) → p, q
  14. Tangent(f', c) → r, s
  15. Tangent(f'_1, c) → t, a
  16. Intersect(g, m) → C
  17. Intersect(h', m) → D
  18. Intersect(g', q) → E
  19. Polygon(C, D, E) → t1, e, c_1, d
  20. Rotate(C, 45°, A) → C'
  21. Rotate(D, 45°, A) → D'
  22. Rotate(E, 45°, A) → E'
  23. Polygon(C', D', E') → t1', e', c', d'
  24. Rotate(C', 45°, A) → C''
  25. Rotate(D', 45°, A) → D''
  26. Rotate(E', 45°, A) → E''
  27. Polygon(C'', D'', E'') → t1'', e'', c'', d''
  28. Rotate(C'', 45°, A) → C'''
  29. Rotate(D'', 45°, A) → D'''
  30. Rotate(E'', 45°, A) → E'''
  31. Polygon(C''', D''', E''') → t1''', e''', c''', d'''
  32. Rotate(C''', 45°, A) → F
  33. Rotate(D''', 45°, A) → G
  34. Rotate(E''', 45°, A) → H
  35. Polygon(F, G, H) → t2, h_1, f_1, g_1
  36. Rotate(F, 45°, A) → F'
  37. Rotate(G, 45°, A) → G'
  38. Rotate(H, 45°, A) → H'
  39. Polygon(F', G', H') → t2', h'_1, f'_2, g'_1
  40. Rotate(F', 45°, A) → F''
  41. Rotate(G', 45°, A) → G''
  42. Rotate(H', 45°, A) → H''
  43. Polygon(F'', G'', H'') → t2'', h''_1, f'', g''
  44. Rotate(F'', 45°, A) → F'''
  45. Rotate(G'', 45°, A) → G'''
  46. Rotate(H'', 45°, A) → H'''
  47. Polygon(F''', G''', H''') → t2''', h''', f''', g'''
  48. Mirror(t1'', c) → t1'''_1
  49. Mirror(t1', c) → t1''_1
  50. Mirror(t1, c) → t1'_1
  51. Mirror(t2''', c) → b
  52. Mirror(t2'', c) → t2'''_1
  53. Mirror(t2', c) → t2''_1
  54. Mirror(t2, c) → t2'_1
  55. Mirror(t1''', c) → o
  56. Intersect(k, r) → I
  57. Intersect(f, i) → J
  58. Intersect(i, p) → K
  59. Polygon(H', I, J, K) → q1, h'_2, i_1, j_1, k_1
  60. Rotate(H', 45°, A) → H''_1
  61. Rotate(I, 45°, A) → I'
  62. Rotate(J, 45°, A) → J'
  63. Rotate(K, 45°, A) → K'
  64. Polygon(H''_1, I', J', K') → q1', h''_2, i', j', k'
  65. Rotate(H''_1, 45°, A) → L
  66. Rotate(I', 45°, A) → I''
  67. Rotate(J', 45°, A) → J''
  68. Rotate(K', 45°, A) → K''
  69. Polygon(L, I'', J'', K'') → q1'', l_1, i'', j'', k''
  70. Rotate(L, 45°, A) → L'
  71. Rotate(I'', 45°, A) → I'''
  72. Rotate(J'', 45°, A) → J'''
  73. Rotate(K'', 45°, A) → K'''
  74. Polygon(L', I''', J''', K''') → q1''', l', i''', j''', k'''
  75. Rotate(L', 45°, A) → L''
  76. Rotate(I''', 45°, A) → M
  77. Rotate(J''', 45°, A) → N
  78. Rotate(K''', 45°, A) → O
  79. Polygon(L'', M, N, O) → q2, l'', m_1, n_1, o_1
  80. Rotate(L'', 45°, A) → L'''
  81. Rotate(M, 45°, A) → M'
  82. Rotate(N, 45°, A) → N'
  83. Rotate(O, 45°, A) → O'
  84. Polygon(L''', M', N', O') → q2', l''', m', n', o'
  85. Rotate(L''', 45°, A) → P
  86. Rotate(M', 45°, A) → M''
  87. Rotate(N', 45°, A) → N''
  88. Rotate(O', 45°, A) → O''
  89. Polygon(P, M'', N'', O'') → q2'', p_1, m'', n'', o''
  90. Rotate(P, 45°, A) → P'
  91. Rotate(M'', 45°, A) → M'''
  92. Rotate(N'', 45°, A) → N'''
  93. Rotate(O'', 45°, A) → O'''
  94. Polygon(P', M''', N''', O''') → q2''', p', m''', n''', o'''
  95. Mirror(q2'', c) → q2'''_1
  96. Mirror(q2', c) → q2''_1
  97. Mirror(q2''', c) → u
  98. Mirror(q1, c) → q1'_1
  99. Mirror(q1', c) → q1''_1
  100. Mirror(q1'', c) → q1'''_1
  101. Mirror(q1''', c) → v
  102. Mirror(q2, c) → q2'_1
  103. Intersect(i, t) → Q
  104. Polygon(Q, G', I, J) → q3, q_1, g'_2, i_2, j_2
  105. Mirror(q3, c) → q3'
  106. Mirror(Q, f) → Q'
  107. Mirror(G', f) → G''_1
  108. Mirror(I, f) → I'_1
  109. Mirror(J, f) → J'_1
  110. Polygon(Q', G''_1, I'_1, J'_1) → q3'_1, q', g''_1, i'_1, j'_1
  111. Mirror(q3', f) → q3''
  112. Mirror(q3', f') → q3''_1
  113. Mirror(q3'', f') → q3'''
  114. Mirror(q3', g) → q3''_2
  115. Mirror(q3'', g) → q3'''_1
  116. Mirror(q3', g') → q3''_3
  117. Mirror(q3'', g') → q3'''_2
  118. Mirror(Q, g') → Q'_1
  119. Mirror(G', g') → G''_2
  120. Mirror(I, g') → I'_2
  121. Mirror(J, g') → J'_2
  122. Polygon(Q'_1, G''_2, I'_2, J'_2) → q3'_2, q'_1, g''_2, i'_2, j'_2
  123. Mirror(Q', g') → Q''
  124. Mirror(G''_1, g') → R
  125. Mirror(I'_1, g') → S
  126. Mirror(J'_1, g') → T
  127. Polygon(Q'', R, S, T) → q4, q'', r_1, s_1, t_1
  128. Mirror(Q, f') → Q'_2
  129. Mirror(G', f') → G''_3
  130. Mirror(I, f') → I'_3
  131. Mirror(J, f') → J'_3
  132. Polygon(Q'_2, G''_3, I'_3, J'_3) → q3'_3, q'_2, g''_3, i'_3, j'_3
  133. Mirror(Q', f') → Q''_1
  134. Mirror(G''_1, f') → U
  135. Mirror(I'_1, f') → V
  136. Mirror(J'_1, f') → W
  137. Polygon(Q''_1, U, V, W) → q5, q''_1, u_1, v_1, w
  138. Mirror(Q''_1, g') → Z
  139. Mirror(U, g') → U'
  140. Mirror(V, g') → V'
  141. Mirror(W, g') → W'
  142. Polygon(Z, U', V', W') → q5', z_1, u', v', w'
  143. Mirror(Q'_2, g') → A_1
  144. Mirror(G''_3, g') → B_1
  145. Mirror(I'_3, g') → C_1
  146. Mirror(J'_3, g') → D_1
  147. Polygon(A_1, B_1, C_1, D_1) → q6, a_1, b_1, c_2, d_1
  148. Intersect(f, n) → E_1
  149. Polygon(A_1, E_1, Z, N) → q7, a_2, e_1, z_2, n_2
  150. Mirror(q7, c) → q7'
  151. Mirror(A_1, f') → F_1
  152. Mirror(E_1, f') → G_1
  153. Mirror(Z, f') → Z'
  154. Mirror(N, f') → N'_1
  155. Polygon(F_1, G_1, Z', N'_1) → q7'_1, f_2, g_2, z', n'_1
  156. Mirror(A_1, g') → H_1
  157. Mirror(E_1, g') → I_1
  158. Mirror(Z, g') → Z'_1
  159. Mirror(N, g') → N'_2
  160. Polygon(H_1, I_1, Z'_1, N'_2) → q7'_2, h_2, i_3, z'_1, n'_2
  161. Mirror(H_1, f') → J_1
  162. Mirror(I_1, f') → K_1
  163. Mirror(Z'_1, f') → L_1
  164. Mirror(N'_2, f') → M_1
  165. Polygon(J_1, K_1, L_1, M_1) → q8, j_3, k_2, l_2, m_2
  166. Mirror(q7', f') → q7''
  167. Mirror(q7'', g') → q7'''
  168. Mirror(q7''', f') → w_1
  169. Intersect(h'', j) → N_1
  170. Polygon(D', A_1, N_1) → t3, n_3, d'_1, a_3
  171. Mirror(t3, c) → t3'
  172. Mirror(D', f) → D''_1
  173. Mirror(A_1, f) → O_1
  174. Mirror(N_1, f) → P_1
  175. Polygon(D''_1, O_1, P_1) → t3'_1, p_2, d''_1, o_2
  176. Mirror(D''_1, f') → Q_1
  177. Mirror(O_1, f') → R_1
  178. Mirror(P_1, f') → S_1
  179. Polygon(Q_1, R_1, S_1) → t4, s_2, q_2, r_2
  180. Mirror(D', f') → D''_2
  181. Mirror(A_1, f') → T_1
  182. Mirror(N_1, f') → U_1
  183. Polygon(D''_2, T_1, U_1) → t3'_2, u_2, d''_2, t_2
  184. Mirror(D''_1, g') → V_1
  185. Mirror(O_1, g') → W_1
  186. Mirror(P_1, g') → Z_1
  187. Polygon(V_1, W_1, Z_1) → t5, z_3, v_2, w_2
  188. Mirror(D', g') → D''_3
  189. Mirror(A_1, g') → A_2
  190. Mirror(N_1, g') → B_2
  191. Polygon(D''_3, A_2, B_2) → t3'_3, b_2, d''_3, a_4
  192. Mirror(V_1, f') → C_2
  193. Mirror(W_1, f') → D_2
  194. Mirror(Z_1, f') → E_2
  195. Polygon(C_2, D_2, E_2) → t5', e_2, c_3, d_2
  196. Mirror(D''_3, f') → F_2
  197. Mirror(A_2, f') → G_2
  198. Mirror(B_2, f') → H_2
  199. Polygon(F_2, G_2, H_2) → t6, h_3, f_3, g_3
  200. Mirror(t3', f) → t3''
  201. Mirror(t3', f') → t3''_1
  202. Mirror(t3'', f') → t3'''
  203. Mirror(f'_1, g') → k_3
  204. Mirror(t3', g') → t3''_2
  205. Mirror(t3'', g') → t3'''_1
  206. Mirror(t3'''_1, f') → b_3
  207. Mirror(t3''_2, f') → d_3
  208. Intersect(g', l) → I_2
  209. Polygon(K''', D', I_2, D, I''', J''') → poly1, k'''_1, d'_2, i_4, d_4, i'''_1, j'''_1
  210. Mirror(poly1, c) → poly1'
  211. Mirror(K''', f) → J_2
  212. Mirror(D', f) → D''_4
  213. Mirror(I_2, f) → K_2
  214. Mirror(D, f) → D'_1
  215. Mirror(I''', f) → L_2
  216. Mirror(J''', f) → M_2
  217. Polygon(J_2, D''_4, K_2, D'_1, L_2, M_2) → poly1'_1, j_4, d''_4, k_4, d'_3, l_3, m_3
  218. Mirror(K''', g) → N_2
  219. Mirror(D', g) → D''_5
  220. Mirror(I_2, g) → O_2
  221. Mirror(D, g) → D'_2
  222. Mirror(I''', g) → P_2
  223. Mirror(J''', g) → Q_2
  224. Polygon(N_2, D''_5, O_2, D'_2, P_2, Q_2) → poly1'_2, n_4, d''_5, o_3, d'_4, p_3, q_3
  225. Mirror(N_2, f) → R_2
  226. Mirror(D''_5, f) → S_2
  227. Mirror(O_2, f) → T_2
  228. Mirror(D'_2, f) → U_2
  229. Mirror(P_2, f) → V_2
  230. Mirror(Q_2, f) → W_2
  231. Polygon(R_2, S_2, T_2, U_2, V_2, W_2) → poly2, r_3, s_3, t_3, u_3, v_3, w_3
  232. Mirror(poly1', f) → poly1''
  233. Mirror(poly1'', g) → poly1'''
  234. Mirror(poly1''', f) → e_3