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Homothety and Translation Symmetry

Homothety and Translation Symmetry
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2018-07-22 06:13
VectorCoordinate GeometryTransformationInequalityGeometric Construction
In collections Homothety, Rotation Symmetry -RO
This example demonstrates the construction of geometric symmetry through homothety and translation operations. It involves core mathematical concepts such as homothety, translation, inequalities, linear equations, limits, random experiments, and tangent lines. The construction utilizes GeoGebra commands like Segment, OrthogonalLine, Point, Intersect, Polygon, AngularBisector, Dilate, Mirror, Vector, and Translate. It is suitable for high school and advanced learners exploring transformation geometry and applying multiple mathematical tools to complex problems.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. OrthogonalLine(A, f) → g
  3. OrthogonalLine(B, f) → h
  4. Point(g) → C
  5. OrthogonalLine(C, g) → i
  6. Intersect(h, i) → D
  7. Polygon(C, D, B, A) → q1, c, d, b, a
  8. Polygon(C, B, D) → t1, d_1, c_1, b_1
  9. AngularBisector(D, C, B) → j
  10. Segment(D, A) → k
  11. Intersect(k, d_1) → E
  12. Point(k) → F
  13. Dilate(C, 9 / 10, F) → C'
  14. Dilate(B, 9 / 10, F) → B'
  15. Dilate(D, 9 / 10, F) → D'
  16. Polygon(C', B', D') → t1', d', c', b'
  17. Dilate(C, 8 / 10, F) → C'_1
  18. Dilate(B, 8 / 10, F) → B'_1
  19. Dilate(D, 8 / 10, F) → D'_1
  20. Polygon(C'_1, B'_1, D'_1) → t1'_1, d'_1, c'_1, b'_1
  21. Dilate(C, 7 / 10, F) → C'_2
  22. Dilate(B, 7 / 10, F) → B'_2
  23. Dilate(D, 7 / 10, F) → D'_2
  24. Polygon(C'_2, B'_2, D'_2) → t1'_2, d'_2, c'_2, b'_2
  25. Dilate(C, 6 / 10, F) → C'_3
  26. Dilate(B, 6 / 10, F) → B'_3
  27. Dilate(D, 6 / 10, F) → D'_3
  28. Polygon(C'_3, B'_3, D'_3) → t1'_3, d'_3, c'_3, b'_3
  29. Dilate(C, 5 / 10, F) → C'_4
  30. Dilate(B, 5 / 10, F) → B'_4
  31. Dilate(D, 5 / 10, F) → D'_4
  32. Polygon(C'_4, B'_4, D'_4) → t1'_4, d'_4, c'_4, b'_4
  33. Dilate(C, 4 / 10, F) → C'_5
  34. Dilate(B, 4 / 10, F) → B'_5
  35. Dilate(D, 4 / 10, F) → D'_5
  36. Polygon(C'_5, B'_5, D'_5) → t1'_5, d'_5, c'_5, b'_5
  37. Dilate(C, 3 / 10, F) → C'_6
  38. Dilate(B, 3 / 10, F) → B'_6
  39. Dilate(D, 3 / 10, F) → D'_6
  40. Polygon(C'_6, B'_6, D'_6) → t1'_6, d'_6, c'_6, b'_6
  41. Dilate(C, 2 / 10, F) → C'_7
  42. Dilate(B, 2 / 10, F) → B'_7
  43. Dilate(D, 2 / 10, F) → D'_7
  44. Polygon(C'_7, B'_7, D'_7) → t1'_7, d'_7, c'_7, b'_7
  45. Mirror(C', E) → C''
  46. Mirror(B', E) → B''
  47. Mirror(D', E) → D''
  48. Polygon(C'', B'', D'') → t1'', d'', c'', b''
  49. Mirror(C', E) → C''_1
  50. Mirror(B', E) → B''_1
  51. Mirror(D', E) → D''_1
  52. Mirror(C'_1, E) → G
  53. Mirror(B'_1, E) → H
  54. Mirror(D'_1, E) → I
  55. Polygon(G, H, I) → t2, i_1, g_1, h_1
  56. Mirror(C'_2, E) → J
  57. Mirror(B'_2, E) → K
  58. Mirror(D'_2, E) → L
  59. Polygon(J, K, L) → t3, l, j_1, k_1
  60. Mirror(C'_3, E) → M
  61. Mirror(B'_3, E) → N
  62. Mirror(D'_3, E) → O
  63. Polygon(M, N, O) → t4, o, m, n
  64. Mirror(C'_4, E) → P
  65. Mirror(B'_4, E) → Q
  66. Mirror(D'_4, E) → R
  67. Polygon(P, Q, R) → t5, r, p, q
  68. Mirror(C'_5, E) → S
  69. Mirror(B'_5, E) → T
  70. Mirror(D'_5, E) → U
  71. Polygon(S, T, U) → t6, u, s, t
  72. Mirror(C'_6, E) → V
  73. Mirror(B'_6, E) → W
  74. Mirror(D'_6, E) → Z
  75. Polygon(V, W, Z) → t7, z_1, v, w
  76. Mirror(C'_7, E) → A_1
  77. Mirror(B'_7, E) → B_1
  78. Mirror(D'_7, E) → C_1
  79. Polygon(A_1, B_1, C_1) → t8, c_2, a_1, b_2
  80. Vector(D_1, E_1) → e
  81. Vector(F_1, G_1) → u_1
  82. Translate(C, e) → C'_8
  83. Translate(D, e) → D'_8
  84. Translate(B, e) → B'_8
  85. Translate(A, e) → A'
  86. Polygon(C'_8, D'_8, B'_8, A') → q1', c'_8, d'_8, b'_8, a'
  87. Translate(C, e) → C'_9
  88. Translate(B, e) → B'_9
  89. Translate(D, e) → D'_9
  90. Polygon(C'_9, B'_9, D'_9) → t1'_8, d'_9, c'_9, b'_9
  91. Translate(C', e) → C''_2
  92. Translate(B', e) → B''_2
  93. Translate(D', e) → D''_2
  94. Polygon(C''_2, B''_2, D''_2) → t1''_1, d''_2, c''_2, b''_2
  95. Translate(C'_1, e) → H_1
  96. Translate(B'_1, e) → I_1
  97. Translate(D'_1, e) → J_1
  98. Polygon(H_1, I_1, J_1) → t9, j_2, h_2, i_2
  99. Translate(C'_2, e) → K_1
  100. Translate(B'_2, e) → L_1
  101. Translate(D'_2, e) → M_1
  102. Polygon(K_1, L_1, M_1) → t10, m_1, k_2, l_1
  103. Translate(C'_3, e) → N_1
  104. Translate(B'_3, e) → O_1
  105. Translate(D'_3, e) → P_1
  106. Polygon(N_1, O_1, P_1) → t11, p_1, n_1, o_1
  107. Translate(C'_4, e) → Q_1
  108. Translate(B'_4, e) → R_1
  109. Translate(D'_4, e) → S_1
  110. Polygon(Q_1, R_1, S_1) → t12, s_1, q_1, r_1
  111. Translate(C'_5, e) → T_1
  112. Translate(B'_5, e) → U_1
  113. Translate(D'_5, e) → V_1
  114. Polygon(T_1, U_1, V_1) → t13, v_1, t_1, u_2
  115. Translate(C'_6, e) → W_1
  116. Translate(B'_6, e) → Z_1
  117. Translate(D'_6, e) → A_2
  118. Polygon(W_1, Z_1, A_2) → t14, a_2, w_1, z_2
  119. Translate(C'_7, e) → B_2
  120. Translate(B'_7, e) → C_2
  121. Translate(D'_7, e) → D_2
  122. Polygon(B_2, C_2, D_2) → t15, d_2, b_3, c_3
  123. Translate(C'', e) → C'''
  124. Translate(B'', e) → B'''
  125. Translate(D'', e) → D'''
  126. Polygon(C''', B''', D''') → t1''', d''', c''', b'''
  127. Translate(G, e) → G'
  128. Translate(H, e) → H'
  129. Translate(I, e) → I'
  130. Polygon(G', H', I') → t2', i', g', h'
  131. Translate(J, e) → J'
  132. Translate(K, e) → K'
  133. Translate(L, e) → L'
  134. Polygon(J', K', L') → t3', l', j', k'
  135. Translate(M, e) → M'
  136. Translate(N, e) → N'
  137. Translate(O, e) → O'
  138. Polygon(M', N', O') → t4', o', m', n'
  139. Translate(P, e) → P'
  140. Translate(Q, e) → Q'
  141. Translate(R, e) → R'
  142. Polygon(P', Q', R') → t5', r', p', q'
  143. Translate(S, e) → S'
  144. Translate(T, e) → T'
  145. Translate(U, e) → U'
  146. Polygon(S', T', U') → t6', u', s', t'
  147. Translate(V, e) → V'
  148. Translate(W, e) → W'
  149. Translate(Z, e) → Z'
  150. Polygon(V', W', Z') → t7', z', v', w'
  151. Translate(A_1, e) → E_2
  152. Translate(B_1, e) → F_2
  153. Translate(C_1, e) → G_2
  154. Polygon(E_2, F_2, G_2) → t8', g_2, e_2, f_2
  155. Translate(C, u_1) → C'_{10}
  156. Translate(D, u_1) → D'_{10}
  157. Translate(B, u_1) → B'_{10}
  158. Translate(A, u_1) → A'_1
  159. Polygon(C'_{10}, D'_{10}, B'_{10}, A'_1) → q1'_1, c'_{10}, d'_{10}, b'_{10}, a'_1
  160. Translate(C, u_1) → C'_{11}
  161. Translate(B, u_1) → B'_{11}
  162. Translate(D, u_1) → D'_{11}
  163. Polygon(C'_{11}, B'_{11}, D'_{11}) → t1'_9, d'_{11}, c'_{11}, b'_{11}
  164. Translate(C', u_1) → C''_3
  165. Translate(B', u_1) → B''_3
  166. Translate(D', u_1) → D''_3
  167. Polygon(C''_3, B''_3, D''_3) → t1''_2, d''_3, c''_3, b''_3
  168. Translate(C'_1, u_1) → H_2
  169. Translate(B'_1, u_1) → I_2
  170. Translate(D'_1, u_1) → J_2
  171. Polygon(H_2, I_2, J_2) → t16, j_3, h_3, i_3
  172. Translate(C'_2, u_1) → K_2
  173. Translate(B'_2, u_1) → L_2
  174. Translate(D'_2, u_1) → M_2
  175. Polygon(K_2, L_2, M_2) → t17, m_2, k_3, l_2
  176. Translate(C'_3, u_1) → N_2
  177. Translate(B'_3, u_1) → O_2
  178. Translate(D'_3, u_1) → P_2
  179. Polygon(N_2, O_2, P_2) → t18, p_2, n_2, o_2
  180. Translate(C'_4, u_1) → Q_2
  181. Translate(B'_4, u_1) → R_2
  182. Translate(D'_4, u_1) → S_2
  183. Polygon(Q_2, R_2, S_2) → t19, s_2, q_2, r_2
  184. Translate(C'_5, u_1) → T_2
  185. Translate(B'_5, u_1) → U_2
  186. Translate(D'_5, u_1) → V_2
  187. Polygon(T_2, U_2, V_2) → t20, v_2, t_2, u_3
  188. Translate(C'_6, u_1) → W_2
  189. Translate(B'_6, u_1) → Z_2
  190. Translate(D'_6, u_1) → A_3
  191. Polygon(W_2, Z_2, A_3) → t21, a_3, w_2, z_3
  192. Translate(C'_7, u_1) → B_3
  193. Translate(B'_7, u_1) → C_3
  194. Translate(D'_7, u_1) → D_3
  195. Polygon(B_3, C_3, D_3) → t22, d_3, b_4, c_4
  196. Translate(C'', u_1) → C'''_1
  197. Translate(B'', u_1) → B'''_1
  198. Translate(D'', u_1) → D'''_1
  199. Polygon(C'''_1, B'''_1, D'''_1) → t1'''_1, d'''_1, c'''_1, b'''_1
  200. Translate(G, u_1) → G'_1
  201. Translate(H, u_1) → H'_1
  202. Translate(I, u_1) → I'_1
  203. Polygon(G'_1, H'_1, I'_1) → t2'_1, i'_1, g'_1, h'_1
  204. Translate(J, u_1) → J'_1
  205. Translate(K, u_1) → K'_1
  206. Translate(L, u_1) → L'_1
  207. Polygon(J'_1, K'_1, L'_1) → t3'_1, l'_1, j'_1, k'_1
  208. Translate(M, u_1) → M'_1
  209. Translate(N, u_1) → N'_1
  210. Translate(O, u_1) → O'_1
  211. Polygon(M'_1, N'_1, O'_1) → t4'_1, o'_1, m'_1, n'_1
  212. Translate(P, u_1) → P'_1
  213. Translate(Q, u_1) → Q'_1
  214. Translate(R, u_1) → R'_1
  215. Polygon(P'_1, Q'_1, R'_1) → t5'_1, r'_1, p'_1, q'_1
  216. Translate(S, u_1) → S'_1
  217. Translate(T, u_1) → T'_1
  218. Translate(U, u_1) → U'_1
  219. Polygon(S'_1, T'_1, U'_1) → t6'_1, u'_1, s'_1, t'_1
  220. Translate(V, u_1) → V'_1
  221. Translate(W, u_1) → W'_1
  222. Translate(Z, u_1) → Z'_1
  223. Polygon(V'_1, W'_1, Z'_1) → t7'_1, z'_1, v'_1, w'_1
  224. Translate(A_1, u_1) → E_3
  225. Translate(B_1, u_1) → F_3
  226. Translate(C_1, u_1) → G_3
  227. Polygon(E_3, F_3, G_3) → t8'_1, g_3, e_3, f_3
  228. Translate(C'_8, u_1) → H_3
  229. Translate(D'_8, u_1) → I_3
  230. Translate(B'_8, u_1) → J_3
  231. Translate(A', u_1) → A''
  232. Polygon(H_3, I_3, J_3, A'') → q1'', h_4, i_4, j_4, a''
  233. Translate(C'_9, u_1) → K_3
  234. Translate(B'_9, u_1) → L_3
  235. Translate(D'_9, u_1) → M_3
  236. Polygon(K_3, L_3, M_3) → t23, m_3, k_4, l_3
  237. Translate(C''_2, u_1) → N_3
  238. Translate(B''_2, u_1) → O_3
  239. Translate(D''_2, u_1) → P_3
  240. Polygon(N_3, O_3, P_3) → t24, p_3, n_3, o_3
  241. Translate(H_1, u_1) → Q_3
  242. Translate(I_1, u_1) → R_3
  243. Translate(J_1, u_1) → S_3
  244. Polygon(Q_3, R_3, S_3) → t9', s_3, q_3, r_3
  245. Translate(K_1, u_1) → T_3
  246. Translate(L_1, u_1) → U_3
  247. Translate(M_1, u_1) → V_3
  248. Polygon(T_3, U_3, V_3) → t10', v_3, t_3, u_4
  249. Translate(N_1, u_1) → W_3
  250. Translate(O_1, u_1) → Z_3
  251. Translate(P_1, u_1) → A_4
  252. Polygon(W_3, Z_3, A_4) → t11', a_4, w_3, z_4
  253. Translate(Q_1, u_1) → B_4
  254. Translate(R_1, u_1) → C_4
  255. Translate(S_1, u_1) → D_4
  256. Polygon(B_4, C_4, D_4) → t12', d_4, b_5, c_5
  257. Translate(T_1, u_1) → E_4
  258. Translate(U_1, u_1) → F_4
  259. Translate(V_1, u_1) → G_4
  260. Polygon(E_4, F_4, G_4) → t13', g_4, e_4, f_4
  261. Translate(W_1, u_1) → H_4
  262. Translate(Z_1, u_1) → I_4
  263. Translate(A_2, u_1) → J_4
  264. Polygon(H_4, I_4, J_4) → t14', j_5, h_5, i_5
  265. Translate(B_2, u_1) → K_4
  266. Translate(C_2, u_1) → L_4
  267. Translate(D_2, u_1) → M_4
  268. Polygon(K_4, L_4, M_4) → t15', m_4, k_5, l_4
  269. Translate(C''', u_1) → N_4
  270. Translate(B''', u_1) → O_4
  271. Translate(D''', u_1) → P_4
  272. Polygon(N_4, O_4, P_4) → t25, p_4, n_4, o_4
  273. Translate(G', u_1) → G''
  274. Translate(H', u_1) → H''
  275. Translate(I', u_1) → I''
  276. Polygon(G'', H'', I'') → t2'', i'', g'', h''
  277. Translate(J', u_1) → J''
  278. Translate(K', u_1) → K''
  279. Translate(L', u_1) → L''
  280. Polygon(J'', K'', L'') → t3'', l'', j'', k''
  281. Translate(M', u_1) → M''
  282. Translate(N', u_1) → N''
  283. Translate(O', u_1) → O''
  284. Polygon(M'', N'', O'') → t4'', o'', m'', n''
  285. Translate(P', u_1) → P''
  286. Translate(Q', u_1) → Q''
  287. Translate(R', u_1) → R''
  288. Polygon(P'', Q'', R'') → t5'', r'', p'', q''
  289. Translate(S', u_1) → S''
  290. Translate(T', u_1) → T''
  291. Translate(U', u_1) → U''
  292. Polygon(S'', T'', U'') → t6'', u'', s'', t''
  293. Translate(V', u_1) → V''
  294. Translate(W', u_1) → W''
  295. Translate(Z', u_1) → Z''
  296. Polygon(V'', W'', Z'') → t7'', z'', v'', w''
  297. Translate(E_2, u_1) → Q_4
  298. Translate(F_2, u_1) → R_4
  299. Translate(G_2, u_1) → S_4
  300. Polygon(Q_4, R_4, S_4) → t8'', s_4, q_4, r_4