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Reflections over a circle 2

Reflections over a circle 2
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2018-04-30 07:27
CircleCoordinate Geometry
In collections Reflections over a circle
This example demonstrates the reflection of points, segments, and polygons over a circle, exploring their trajectories and properties. It involves core mathematical concepts such as circles, reflections, coordinate geometry, transformations, continuity, parabolas, congruence, and linear regression. The construction utilizes commands like Circle, Line, OrthogonalLine, Rotate, Tangent, Intersect, Polygon, and Mirror. Suitable for high school and advanced learners studying geometric transformations, symmetry, and data analysis.

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
  235. Intersect(g', s_3) → Z_2
  236. Intersect(h'_1, f'_2) → A_3
  237. Intersect(f_3, g_3) → B_3
  238. Polygon(Z_2, A_3, B_3) → t7, b_4, z_4, a_5
  239. Mirror(t7, c) → t7'
  240. Mirror(Z_2, g') → C_3
  241. Mirror(A_3, g') → D_3
  242. Mirror(B_3, g') → E_3
  243. Polygon(C_3, D_3, E_3) → t7'_1, e_4, c_4, d_5
  244. Mirror(C_3, g) → F_3
  245. Mirror(D_3, g) → G_3
  246. Mirror(E_3, g) → H_3
  247. Polygon(F_3, G_3, H_3) → t8, h_4, f_4, g_4
  248. Mirror(Z_2, g) → I_3
  249. Mirror(A_3, g) → J_3
  250. Mirror(B_3, g) → K_3
  251. Polygon(I_3, J_3, K_3) → t7'_2, k_5, i_5, j_5
  252. Mirror(I_3, f) → L_3
  253. Mirror(J_3, f) → M_3
  254. Mirror(K_3, f) → N_3
  255. Polygon(L_3, M_3, N_3) → t9, n_5, l_4, m_4
  256. Mirror(F_3, f) → O_3
  257. Mirror(G_3, f) → P_3
  258. Mirror(H_3, f) → Q_3
  259. Polygon(O_3, P_3, Q_3) → t8', q_4, o_4, p_4
  260. Mirror(O_3, g) → R_3
  261. Mirror(P_3, g) → S_3
  262. Mirror(Q_3, g) → T_3
  263. Polygon(R_3, S_3, T_3) → t8'', t_4, r_4, s_4
  264. Mirror(L_3, f') → U_3
  265. Mirror(M_3, f') → V_3
  266. Mirror(N_3, f') → W_3
  267. Mirror(L_3, g) → Z_3
  268. Mirror(M_3, g) → A_4
  269. Mirror(N_3, g) → B_4
  270. Polygon(Z_3, A_4, B_4) → t9'_1, b_5, z_5, a_6
  271. Mirror(t7', g') → t7''
  272. Mirror(t7'', g) → t7'''
  273. Mirror(t7', g) → t7''_1
  274. Mirror(t7''_1, f) → u_4
  275. Mirror(t7''', f) → v_4
  276. Mirror(v_4, g) → w_4
  277. Mirror(u_4, f') → c_5
  278. Mirror(u_4, g) → e_5
  279. Intersect(q_1, j_2) → C_4
  280. Intersect(f_3, g_3) → D_4
  281. Intersect(f, j_3) → E_4
  282. Polygon(C_4, D_4, E_4) → t10, e_6, c_6, d_6
  283. Mirror(t10, c) → t10'
  284. Mirror(C_4, g') → F_4
  285. Mirror(D_4, g') → G_4
  286. Mirror(E_4, g') → H_4
  287. Polygon(F_4, G_4, H_4) → t10'_1, h_5, f_5, g_5
  288. Mirror(F_4, g) → I_4
  289. Mirror(G_4, g) → J_4
  290. Mirror(H_4, g) → K_4
  291. Polygon(I_4, J_4, K_4) → t11, k_6, i_6, j_6
  292. Mirror(I_4, f') → L_4
  293. Mirror(J_4, f') → M_4
  294. Mirror(K_4, f') → N_4
  295. Polygon(L_4, M_4, N_4) → t11', n_6, l_5, m_5
  296. Mirror(F_4, f') → O_4
  297. Mirror(G_4, f') → P_4
  298. Mirror(H_4, f') → Q_4
  299. Polygon(O_4, P_4, Q_4) → t12, q_5, o_5, p_5
  300. Mirror(O_4, g') → R_4
  301. Mirror(P_4, g') → S_4
  302. Mirror(Q_4, g') → T_4
  303. Polygon(R_4, S_4, T_4) → t12', t_5, r_5, s_5
  304. Mirror(L_4, g') → U_4
  305. Mirror(M_4, g') → V_4
  306. Mirror(N_4, g') → W_4
  307. Polygon(U_4, V_4, W_4) → t11'', w_5, u_5, v_5
  308. Mirror(C_4, f) → Z_4
  309. Mirror(D_4, f) → A_5
  310. Mirror(E_4, f) → B_5
  311. Polygon(Z_4, A_5, B_5) → t10'_2, b_6, z_6, a_7
  312. Mirror(t10', f) → t10''
  313. Mirror(t10', g') → t10''_1
  314. Mirror(t10'', g') → t10'''
  315. Mirror(t10''_1, f') → f_6
  316. Mirror(t10''', f') → g_6
  317. Mirror(g_6, g') → h_6
  318. Mirror(f_6, g') → l_6