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Symmetry Lines in the Letter X: Construction & Verification

Symmetry Lines in the Letter X: Construction & Verification
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2020-12-24 17:16
CirclePolygonAngleTransformationGeometric Construction
This example demonstrates how to identify and visualize the lines of symmetry of a composite “X” figure built from geometric segments and circular arcs. It highlights the matching of corresponding points under reflection, making the symmetry visually verifiable rather than assumed. Key mathematical ideas include symmetry axes, mirror reflection (Mirror), midpoint-based construction (Midpoint, LineBisector), angle construction/measurement (Angle), and using circles or circular sectors (Circle, CircleSector) as auxiliary curved parts. Construction workflow: Polygon and Segment build the main polygonal components; Intersect finds crucial intersection points; Point and Line set reference elements; Circle and CircleSector generate arc/curved features; Mirror (together with Rotate and Translate when needed) maps the figure onto itself to confirm invariance; Midpoint and LineBisector provide baseline lines linked to the symmetry; Textfield/Text are used to label symmetry-axis results. Suitable for middle school to early university geometry lessons and for exploring transformation geometry and geometric construction.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → mnogokut1, f, g, h, i, C, D
  2. Segment(B, D) → k
  3. Segment(A, C) → j
  4. Intersect(j, k) → E
  5. Circle(E, B) → c
  6. Point(c) → F
  7. Line(F, E) → l
  8. Intersect(c, l, 2) → G
  9. CircleSector(E, G, F) → polukrug
  10. Point(l) → H
  11. Point(l) → I
  12. Segment(H, I) → m
  13. Polygon(J, J', 4) → mnogokut2, n, p, q, r, K, L
  14. Segment(J, K) → s
  15. Segment(J', L) → t
  16. Intersect(s, t) → M
  17. Line(M, l) → a
  18. Point(a) → N
  19. Point(a) → O
  20. Segment(N, O) → b
  21. Polygon(A, B, T_1) → mnogokut3, t_1, a_1, b_1
  22. Mirror(A, l) → A'
  23. Mirror(B, l) → B'
  24. Mirror(T_1, l) → P
  25. Polygon(A', B', P) → mnogokut3', p_1, a', b'
  26. Polygon(J, J', T_2) → mnogokut3_1, t_2, j_1, j'
  27. Midpoint(Segment[D, C]) → T_3
  28. Polygon(A, B, T_3) → mnogokut4, t_3, a_2, b_2
  29. Midpoint(Segment[L, K]) → T_4
  30. Polygon(J, J', T_4) → mnogokut4_1, t_4, j_2, j'_1
  31. Mirror(A, l) → A'_1
  32. Mirror(B, l) → B'_1
  33. Mirror(T_3, l) → Q
  34. Polygon(A'_1, B'_1, Q) → mnogokut4', q_1, a'_1, b'_1
  35. Polygon(A, B, 3) → mnogokut5, d, e, f_1, R
  36. LineBisector(d) → g_1
  37. LineBisector(e) → h_1
  38. Intersect(g_1, h_1) → S
  39. Circle(S, B) → c_1
  40. Point(c_1) → T
  41. Line(T, S) → i_1
  42. Intersect(c_1, i_1, 2) → U
  43. CircleSector(S, U, T) → d_1
  44. Point(i_1) → V
  45. Point(i_1) → W
  46. Segment(V, W) → k_1
  47. Mirror(mnogokut5, i_1) → mnogokut5'
  48. Polygon(J, J', 3) → mnogokut6, l_1, m_1, r_1, Z
  49. LineBisector(l_1) → s_1
  50. LineBisector(m_1) → e_1
  51. Intersect(s_1, e_1) → A_1
  52. Line(A_1, k_1) → f_2
  53. Point(f_2) → B_1
  54. Point(f_2) → C_1
  55. Segment(C_1, B_1) → g_2
  56. Polygon(A, B, 4) → mnogokut7, h_2, i_2, k_2, l_2, D_1, E_1
  57. Mirror(mnogokut7, l) → mnogokut7'
  58. Polygon(J, J', 4) → mnogokut7_1, m_2, n_2, p_2, q_2, F_1, G_1
  59. Line(A, I_1) → r_2
  60. Circle(A, Segment[A, H_1]) → c_2
  61. Intersect(c_2, r_2, 2) → J_1
  62. Circle(J_1, Segment[A, H_1]) → d_2
  63. Circle(H_1, Segment[A, H_1]) → e_2
  64. Intersect(d_2, e_2, 1) → K_1
  65. Polygon(A, H_1, K_1, J_1) → mnogokut8, a_3, h_3, k_3, j_3
  66. Segment(A, K_1) → s_2
  67. Segment(H_1, J_1) → f_3
  68. Intersect(s_2, f_3) → L_1
  69. Circle(L_1, K_1) → c_3
  70. Point(c_3) → M_1
  71. Line(M_1, L_1) → g_3
  72. Intersect(c_3, g_3, 2) → N_1
  73. CircleSector(L_1, N_1, M_1) → d_3
  74. Mirror(A, g_3) → A'_2
  75. Mirror(H_1, g_3) → O_1
  76. Mirror(K_1, g_3) → P_1
  77. Mirror(J_1, g_3) → Q_1
  78. Polygon(A'_2, O_1, P_1, Q_1) → mnogokut8', a'_2, o_1, p_3, q_3
  79. Point(g_3) → R_1
  80. Point(g_3) → S_1
  81. Segment(S_1, R_1) → i_3
  82. Line(J, r_2) → l_3
  83. Circle(J, Segment[J, U_1]) → e_3
  84. Intersect(e_3, l_3, 2) → V_1
  85. Circle(V_1, Segment[J, U_1]) → r_3
  86. Circle(U_1, Segment[J, U_1]) → s_3
  87. Intersect(r_3, s_3, 1) → W_1
  88. Polygon(J, U_1, W_1, V_1) → mnogokut9, j_4, u_1, w_1, v_1
  89. Segment(J, W_1) → m_3
  90. Segment(U_1, V_1) → n_3
  91. Intersect(m_3, n_3) → Z_1
  92. Line(Z_1, g_3) → b_3
  93. Point(b_3) → A_2
  94. Point(b_3) → B_2
  95. Segment(B_2, A_2) → f_4
  96. Circle(A, Segment[B, C_2]) → c_4
  97. Circle(C_2, Segment[A, B]) → d_4
  98. Intersect(c_4, d_4, 1) → D_2
  99. Polygon(A, B, C_2, D_2) → mnogokut10, a_4, b_4, c_5, d_5
  100. Segment(A, C_2) → g_4
  101. Segment(B, D_2) → h_4
  102. Intersect(g_4, h_4) → E_2
  103. Circle(E_2, B) → e_4
  104. Point(e_4) → F_2
  105. Line(F_2, E_2) → i_4
  106. Intersect(e_4, i_4, 2) → G_2
  107. CircleSector(E_2, G_2, F_2) → k_4
  108. Mirror(A, i_4) → A'_3
  109. Mirror(B, i_4) → B'_2
  110. Mirror(C_2, i_4) → H_2
  111. Mirror(D_2, i_4) → I_2
  112. Polygon(A'_3, B'_2, H_2, I_2) → mnogokut10', a'_3, b'_2, h_5, i_5
  113. Point(i_4) → J_2
  114. Point(i_4) → K_2
  115. Circle(J, Segment[J', L_2]) → p_4
  116. Circle(L_2, Segment[J, J']) → q_4
  117. Intersect(p_4, q_4, 1) → M_2
  118. Polygon(J, J', L_2, M_2) → mnogokut10_1, j_5, j'_2, l_4, m_4
  119. Segment(J, L_2) → n_4
  120. Segment(J', M_2) → r_4
  121. Intersect(n_4, r_4) → N_2
  122. Line(N_2, i_4) → s_4
  123. Point(s_4) → O_2
  124. Point(s_4) → P_2
  125. Segment(K_2, J_2) → f_5
  126. Segment(P_2, O_2) → g_5
  127. Polygon(A, B, R_2, Q_2) → mnogokut11, a_5, b_5, r_5, q_5
  128. Mirror(A, l) → A'_4
  129. Mirror(B, l) → B'_3
  130. Mirror(R_2, l) → S_2
  131. Mirror(Q_2, l) → U_2
  132. Polygon(A'_4, B'_3, S_2, U_2) → mnogokut11', a'_4, b'_3, s_5, u_2
  133. Polygon(J, J', W_2, V_2) → mnogokut12, j_6, j'_3, w_2, v_2
  134. Circle(A, 3) → e_5
  135. Intersect(e_5, r_2, 2) → Z_2
  136. Circle(Z_2, Segment[A, B]) → k_5
  137. Circle(B, Segment[A, Z_2]) → p_5
  138. Intersect(k_5, p_5, 1) → A_3
  139. Polygon(A, B, A_3, Z_2) → mnogokut13, a_6, b_6, a_7, z_2
  140. Segment(A, A_3) → l_5
  141. Segment(B, Z_2) → m_5
  142. Intersect(l_5, m_5) → B_3
  143. Circle(B_3, A_3) → t_5
  144. Point(t_5) → C_3
  145. Line(C_3, B_3) → n_5
  146. Intersect(t_5, n_5, 2) → D_3
  147. Line(J, r_2) → f_6
  148. CircleSector(B_3, D_3, C_3) → c_6
  149. Mirror(A, n_5) → A'_5
  150. Mirror(B, n_5) → B'_4
  151. Mirror(A_3, n_5) → E_3
  152. Mirror(Z_2, n_5) → F_3
  153. Polygon(A'_5, B'_4, E_3, F_3) → mnogokut13', a'_5, b'_4, e_6, f_7
  154. Point(n_5) → G_3
  155. Point(n_5) → H_3
  156. Circle(J, 3) → d_6
  157. Intersect(d_6, f_6, 2) → I_3
  158. Circle(I_3, Segment[J, J']) → g_6
  159. Circle(J', Segment[J, I_3]) → h_6
  160. Intersect(g_6, h_6, 1) → J_3
  161. Polygon(J, J', J_3, I_3) → mnogokut13_1, j_7, j'_4, j_8, i_6
  162. Segment(J, J_3) → k_6
  163. Segment(J', I_3) → l_6
  164. Intersect(k_6, l_6) → K_3
  165. Line(K_3, n_5) → m_6
  166. Point(m_6) → L_3
  167. Point(m_6) → M_3
  168. Segment(H_3, G_3) → n_6
  169. Segment(M_3, L_3) → p_6
  170. Line(E, yAxis) → q_6
  171. Intersect(c, q_6, 1) → N_3
  172. Rotate(N_3, 36°, E) → O_3
  173. Angle(N_3, E, O_3) → α
  174. Rotate(N_3, (-(36°)), E) → P_3
  175. Angle(P_3, E, N_3) → β
  176. Circle(O_3, Segment[P_3, O_3]) → r_6
  177. Intersect(c, r_6, 1) → Q_3
  178. Circle(Q_3, Segment[O_3, Q_3]) → s_6
  179. Intersect(c, s_6, 1) → R_3
  180. Circle(R_3, Segment[Q_3, R_3]) → t_6
  181. Intersect(c, t_6, 1) → S_3
  182. Polygon(P_3, O_3, Q_3, R_3, S_3) → mnogokut14, p_7, o_3, q_7, r_7, s_7
  183. Mirror(P_3, l) → U_3
  184. Mirror(O_3, l) → V_3
  185. Mirror(Q_3, l) → W_3
  186. Mirror(R_3, l) → Z_3
  187. Mirror(S_3, l) → A_4
  188. Polygon(U_3, V_3, W_3, Z_3, A_4) → mnogokut14', u_3, v_3, w_3, z_3, a_8
  189. Translate(P_3, Vector[A, J]) → B_4
  190. Translate(O_3, Vector[A, J]) → C_4
  191. Translate(Q_3, Vector[A, J]) → D_4
  192. Translate(R_3, Vector[A, J]) → E_4
  193. Translate(S_3, Vector[A, J]) → F_4
  194. Polygon(B_4, C_4, D_4, E_4, F_4) → mnogokut14'_1, b_7, c_7, d_7, e_7, f_8
  195. Rotate(N_3, 30°, E) → G_4
  196. Angle(N_3, E, G_4) → γ
  197. Rotate(N_3, (-(30°)), E) → H_4
  198. Angle(H_4, E, N_3) → δ
  199. Circle(G_4, Segment[H_4, G_4]) → g_7
  200. Intersect(c, g_7, 1) → I_4
  201. Circle(I_4, Segment[G_4, I_4]) → h_7
  202. Intersect(c, h_7, 1) → J_4
  203. Circle(J_4, Segment[I_4, J_4]) → k_7
  204. Intersect(c, k_7, 1) → K_4
  205. Circle(K_4, Segment[J_4, K_4]) → t_7
  206. Intersect(c, t_7, 1) → L_4
  207. Polygon(H_4, G_4, I_4, J_4, K_4, L_4) → mnogokut15, h_8, g_8, i_7, j_9, k_8, l_7
  208. Mirror(H_4, l) → M_4
  209. Mirror(G_4, l) → N_4
  210. Mirror(I_4, l) → O_4
  211. Mirror(J_4, l) → P_4
  212. Mirror(K_4, l) → Q_4
  213. Mirror(L_4, l) → R_4
  214. Polygon(M_4, N_4, O_4, P_4, Q_4, R_4) → mnogokut15', m_7, n_7, o_4, p_8, q_8, r_8
  215. Translate(H_4, Vector[A, J]) → S_4
  216. Translate(G_4, Vector[A, J]) → U_4
  217. Translate(I_4, Vector[A, J]) → V_4
  218. Translate(J_4, Vector[A, J]) → W_4
  219. Translate(K_4, Vector[A, J]) → Z_4
  220. Translate(L_4, Vector[A, J]) → A_5
  221. Polygon(S_4, U_4, V_4, W_4, Z_4, A_5) → mnogokut15'_1, s_8, u_4, v_4, w_4, z_4, a_9
  222. Polygon(B_5, C_5, D_5, L_5, M_5, E_5, I_5, K_5, H_5, F_5, J_5, G_5) → mnogokut16, b_8, g_9, d_8, l_8, c_8, m_8, e_8, i_8, k_9, h_9, f_9, j_{10}
  223. Mirror(B_5, l) → N_5
  224. Mirror(C_5, l) → O_5
  225. Mirror(M_5, l) → P_5
  226. Mirror(E_5, l) → Q_5
  227. Mirror(I_5, l) → R_5
  228. Mirror(K_5, l) → S_5
  229. Mirror(H_5, l) → T_5
  230. Mirror(F_5, l) → U_5
  231. Mirror(J_5, l) → V_5
  232. Mirror(G_5, l) → W_5
  233. Mirror(D_5, l) → Z_5
  234. Mirror(L_5, l) → A_6
  235. Polygon(N_5, O_5, Z_5, A_6, P_5, Q_5, R_5, S_5, T_5, U_5, V_5, W_5) → mnogokut16', n_8, w_5, z_5, a_{10}, o_5, p_9, q_9, r_9, s_9, t_8, u_5, v_5
  236. Translate(B_5, Vector[A, J]) → B_6
  237. Translate(C_5, Vector[A, J]) → C_6
  238. Translate(M_5, Vector[A, J]) → D_6
  239. Translate(E_5, Vector[A, J]) → E_6
  240. Translate(I_5, Vector[A, J]) → F_6
  241. Translate(K_5, Vector[A, J]) → G_6
  242. Translate(H_5, Vector[A, J]) → H_6
  243. Translate(F_5, Vector[A, J]) → I_6
  244. Translate(J_5, Vector[A, J]) → J_6
  245. Translate(G_5, Vector[A, J]) → K_6
  246. Translate(D_5, Vector[A, J]) → L_6
  247. Translate(L_5, Vector[A, J]) → M_6
  248. Polygon(B_6, C_6, L_6, M_6, D_6, E_6, F_6, G_6, H_6, I_6, J_6, K_6) → mnogokut16'_1, b_9, k_{10}, l_9, m_9, c_9, d_9, e_9, f_{10}, g_{10}, h_{10}, i_9, j_{11}
  249. Polygon(A, P_6, N_6, D, S_6, U_6, Q_6, B, C, O_6, T_6, R_6) → mnogokut17, a_{11}, r_{10}, n_9, d_{10}, p_{10}, s_{10}, u_6, q_{10}, b_{10}, c_{10}, o_6, t_9
  250. Mirror(A, l) → A'_6
  251. Mirror(P_6, l) → V_6
  252. Mirror(S_6, l) → W_6
  253. Mirror(U_6, l) → Z_6
  254. Mirror(Q_6, l) → A_7
  255. Mirror(B, l) → B'_5
  256. Mirror(C, l) → C'
  257. Mirror(O_6, l) → B_7
  258. Mirror(T_6, l) → C_7
  259. Mirror(R_6, l) → D_7
  260. Mirror(N_6, l) → E_7
  261. Mirror(D, l) → D'
  262. Polygon(A'_6, V_6, E_7, D', W_6, Z_6, A_7, B'_5, C', B_7, C_7, D_7) → mnogokut17', a'_6, d_{11}, e_{10}, d', v_6, w_6, z_6, a_{12}, b'_5, c', b_{11}, c_{11}
  263. Translate(A, Vector[A, J]) → A'_7
  264. Translate(P_6, Vector[A, J]) → F_7
  265. Translate(S_6, Vector[A, J]) → G_7
  266. Translate(U_6, Vector[A, J]) → H_7
  267. Translate(Q_6, Vector[A, J]) → I_7
  268. Translate(B, Vector[A, J]) → B'_6
  269. Translate(C, Vector[A, J]) → C'_1
  270. Translate(O_6, Vector[A, J]) → J_7
  271. Translate(T_6, Vector[A, J]) → K_7
  272. Translate(R_6, Vector[A, J]) → L_7
  273. Translate(N_6, Vector[A, J]) → M_7
  274. Translate(D, Vector[A, J]) → D'_1
  275. Polygon(A'_7, F_7, M_7, D'_1, G_7, H_7, I_7, B'_6, C'_1, J_7, K_7, L_7) → mnogokut17'_1, a'_7, l_{10}, m_{10}, d'_1, f_{11}, g_{11}, h_{11}, i_{10}, b'_6, c'_1, j_{12}, k_{11}
  276. Polygon(R_7, S_7, Q_7, O_7, C, D, N_7, P_7) → mnogokut18, r_{11}, s_{11}, q_{11}, o_7, c_{12}, d_{12}, n_{10}, p_{11}
  277. Mirror(R_7, l) → T_7
  278. Mirror(S_7, l) → U_7
  279. Mirror(Q_7, l) → V_7
  280. Mirror(O_7, l) → W_7
  281. Mirror(C, l) → C'_2
  282. Mirror(D, l) → D'_2
  283. Mirror(N_7, l) → Z_7
  284. Mirror(P_7, l) → A_8
  285. Polygon(T_7, U_7, V_7, W_7, C'_2, D'_2, Z_7, A_8) → mnogokut18', t_{10}, u_7, v_7, w_7, c'_2, d'_2, z_7, a_{13}
  286. Translate(R_7, Vector[A, J]) → B_8
  287. Translate(S_7, Vector[A, J]) → C_8
  288. Translate(Q_7, Vector[A, J]) → D_8
  289. Translate(O_7, Vector[A, J]) → E_8
  290. Translate(C, Vector[A, J]) → C'_3
  291. Translate(D, Vector[A, J]) → D'_3
  292. Translate(N_7, Vector[A, J]) → F_8
  293. Translate(P_7, Vector[A, J]) → G_8
  294. Polygon(B_8, C_8, D_8, E_8, C'_3, D'_3, F_8, G_8) → mnogokut18'_1, b_{12}, c_{13}, d_{13}, e_{11}, c'_3, d'_3, f_{12}, g_{12}
  295. Polygon(D, H_8, A, O_8, M_8, I_8, C, N_8, B, K_8, L_8, J_8) → mnogokut19, d_{14}, j_{13}, a_{14}, o_8, h_{12}, m_{11}, i_{11}, c_{14}, n_{11}, b_{13}, k_{12}, l_{11}
  296. Mirror(D, l) → D'_4
  297. Mirror(H_8, l) → P_8
  298. Mirror(M_8, l) → Q_8
  299. Mirror(I_8, l) → R_8
  300. Mirror(C, l) → C'_4
  301. Mirror(N_8, l) → S_8
  302. Mirror(B, l) → B'_7
  303. Mirror(K_8, l) → T_8
  304. Mirror(L_8, l) → U_8
  305. Mirror(J_8, l) → V_8
  306. Mirror(A, l) → A'_8
  307. Mirror(O_8, l) → W_8
  308. Polygon(D'_4, P_8, A'_8, W_8, Q_8, R_8, C'_4, S_8, B'_7, T_8, U_8, V_8) → mnogokut19', d'_4, v_8, a'_8, w_8, p_{12}, q_{12}, r_{12}, c'_4, s_{12}, b'_7, t_{11}, u_8
  309. Translate(D, Vector[A, J]) → D'_5
  310. Translate(H_8, Vector[A, J]) → Z_8
  311. Translate(M_8, Vector[A, J]) → A_9
  312. Translate(I_8, Vector[A, J]) → B_9
  313. Translate(C, Vector[A, J]) → C'_5
  314. Translate(N_8, Vector[A, J]) → C_9
  315. Translate(B, Vector[A, J]) → B'_8
  316. Translate(K_8, Vector[A, J]) → D_9
  317. Translate(L_8, Vector[A, J]) → E_9
  318. Translate(J_8, Vector[A, J]) → F_9
  319. Translate(A, Vector[A, J]) → A'_9
  320. Translate(O_8, Vector[A, J]) → G_9
  321. Polygon(D'_5, Z_8, A'_9, G_9, A_9, B_9, C'_5, C_9, B'_8, D_9, E_9, F_9) → mnogokut19'_1, d'_5, f_{13}, a'_9, g_{13}, z_8, a_{15}, b_{14}, c'_5, c_{15}, b'_8, d_{15}, e_{12}
  322. Textfield(brOsiUpisano) → txtPolje1
  323. Element(brojOsiSimetrije, 14) → brOsi

AI Drawing Prompt Reference

Draw a composite X-shaped figure made of squares and circular arcs. Start with a large square, draw its diagonals, and use their intersection as the center of a circle passing through a square vertex. Pick a point on the circle, draw a line through it and the center, intersect the circle again, and form a semicircle with these two points and the center. Then draw a smaller square, connect its diagonals, and use the intersection with the previous semicircle's center to define a symmetry axis. Place two points on this axis and draw a segment, then construct a triangle that is mirrored across the axis. Finally, draw an equilateral triangle, construct perpendicular bisectors of two sides, use their intersection as a circle center, draw a circle through a vertex, pick a point on it, draw a line through the center, form a semicircle, and mirror the triangle across this line. Show all symmetry axes clearly and ensure corresponding parts are visibly reflected.
The prompt above was compiled from the original drawing commands and is for reference only.