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Adding Fractions with Unlike Denominators (Fraction Tiles)

Adding Fractions with Unlike Denominators (Fraction Tiles)
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2018-08-12 12:52
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① This interactive GeoGebra example uses fraction tiles to visually demonstrate how to add fractions with unlike denominators, making the process of finding a common denominator tangible. ② Core math concepts include fractions, addition of unlike denominators, equivalent fractions, and the visual representation of unit fractions. ③ Construction highlights: The model is built using GeoGebra commands such as Segment, OrthogonalLine, Circle, Intersect, Polygon, Point, and Line, creating objects like A, B, C, poly1, poly2, and auxiliary lines to form adjustable fraction bars that can be combined. ④ Suitable for upper elementary to middle school students learning fraction operations, and useful for teachers in classroom demonstrations or for student-led exploration of fraction addition.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. OrthogonalLine(B, a) → c
  3. Circle(B, A) → d
  4. Intersect(d, c, 2) → C
  5. OrthogonalLine(A, a) → e
  6. Circle(A, B) → f
  7. Intersect(f, e, 2) → D
  8. Intersect(d, c, 2) → E
  9. Intersect(d, c, 2) → F
  10. Intersect(f, e, 2) → H
  11. Polygon(A, B, C, D) → poly1, g, h, i, j
  12. Segment(G, I) → b
  13. OrthogonalLine(I, b) → k
  14. Point(k) → J
  15. Line(J, b) → l
  16. OrthogonalLine(G, b) → m
  17. Intersect(l, m) → K
  18. Polygon(G, I, J, K) → poly2, g_1, i_1, j_1, k_1
  19. Segment(R, S) → s
  20. OrthogonalLine(S, s) → t
  21. Point(t) → T
  22. Line(T, s) → a_1
  23. OrthogonalLine(R, s) → b_1
  24. Intersect(b_1, a_1) → U
  25. Polygon(R, S, T, U) → poly3, r_1, s_1, t_1, u
  26. Segment(V, W) → c_1
  27. OrthogonalLine(V, c_1) → f_1
  28. Segment(B_1, C_1) → h_1
  29. OrthogonalLine(C_1, h_1) → l_1
  30. Point(l_1) → D_1
  31. Line(D_1, h_1) → m_1
  32. OrthogonalLine(B_1, h_1) → n_1
  33. Intersect(n_1, m_1) → E_1
  34. Polygon(B_1, C_1, D_1, E_1) → poly5, b_2, c_2, d_2, e_2
  35. Segment(B_2, C_2) → h_2
  36. OrthogonalLine(C_2, h_2) → l_2
  37. Point(l_2) → D_2
  38. Line(D_2, h_2) → m_2
  39. OrthogonalLine(B_2, h_2) → n_2
  40. Intersect(n_2, m_2) → E_2
  41. Polygon(B_2, C_2, D_2, E_2) → poly5_1, b_3, c_3, d_3, e_3
  42. Segment(B_3, C_3) → h_3
  43. OrthogonalLine(C_3, h_3) → l_3
  44. Point(l_3) → D_3
  45. Line(D_3, h_3) → m_3
  46. OrthogonalLine(B_3, h_3) → n_3
  47. Intersect(n_3, m_3) → E_3
  48. Polygon(B_3, C_3, D_3, E_3) → poly5_2, b_4, c_4, d_4, e_4
  49. Segment(V_1, W_1) → c_5
  50. OrthogonalLine(W_1, c_5) → d_5
  51. Point(d_5) → Z_1
  52. Line(Z_1, c_5) → e_5
  53. OrthogonalLine(V_1, c_5) → f_2
  54. Intersect(e_5, f_2) → A_2
  55. Segment(V_2, W_2) → c_6
  56. OrthogonalLine(W_2, c_6) → d_6
  57. Point(d_6) → Z_2
  58. Line(Z_2, c_6) → e_6
  59. OrthogonalLine(V_2, c_6) → f_3
  60. Intersect(e_6, f_3) → A_3
  61. Segment(V_3, W_3) → c_7
  62. OrthogonalLine(W_3, c_7) → d_7
  63. Point(d_7) → Z_3
  64. Line(Z_3, c_7) → e_7
  65. OrthogonalLine(V_3, c_7) → f_4
  66. Intersect(e_7, f_4) → A_4
  67. Polygon(V_3, W_3, Z_3, A_4) → poly4_1, v_1, w_1, z_1, a_3
  68. Segment(V_4, W_4) → c_8
  69. OrthogonalLine(W_4, c_8) → d_8
  70. Point(d_8) → Z_4
  71. Line(Z_4, c_8) → e_8
  72. OrthogonalLine(V_4, c_8) → f_5
  73. Intersect(e_8, f_5) → A_5
  74. Polygon(V_4, W_4, Z_4, A_5) → poly4_2, v_2, w_2, z_2, a_4
  75. Segment(V_5, W_5) → c_9
  76. OrthogonalLine(W_5, c_9) → d_9
  77. Point(d_9) → Z_5
  78. Line(Z_5, c_9) → e_9
  79. OrthogonalLine(V_5, c_9) → f_6
  80. Intersect(e_9, f_6) → A_6
  81. Polygon(V_5, W_5, Z_5, A_6) → poly4_3, v_3, w_3, z_3, a_5
  82. Segment(V_6, W_6) → c_{10}
  83. OrthogonalLine(W_6, c_{10}) → d_{10}
  84. Point(d_{10}) → Z_6
  85. Line(Z_6, c_{10}) → e_{10}
  86. OrthogonalLine(V_6, c_{10}) → f_7
  87. Intersect(e_{10}, f_7) → A_7
  88. Polygon(V_6, W_6, Z_6, A_7) → poly4_4, v_4, w_4, z_4, a_6
  89. Segment(A_8, B_4) → a_7
  90. OrthogonalLine(B_4, a_7) → c_{11}
  91. Circle(B_4, A_8) → d_{11}
  92. Intersect(d_{11}, c_{11}, 2) → C_4
  93. OrthogonalLine(A_8, a_7) → e_{11}
  94. Circle(A_8, B_4) → f_8
  95. Intersect(f_8, e_{11}, 2) → D_4
  96. Polygon(A_8, B_4, C_4, D_4) → poly1_1, g_2, h_4, i_2, j_2
  97. Segment(A_9, B_5) → a_8
  98. OrthogonalLine(B_5, a_8) → c_{12}
  99. Circle(B_5, A_9) → d_{12}
  100. Intersect(d_{12}, c_{12}, 2) → C_5
  101. OrthogonalLine(A_9, a_8) → e_{12}
  102. Circle(A_9, B_5) → f_9
  103. Intersect(f_9, e_{12}, 2) → D_5
  104. Polygon(A_9, B_5, C_5, D_5) → poly1_2, g_3, h_5, i_3, j_3
  105. Segment(A_{10}, B_6) → a_9
  106. OrthogonalLine(B_6, a_9) → c_{13}
  107. Circle(B_6, A_{10}) → d_{13}
  108. Intersect(d_{13}, c_{13}, 2) → C_6
  109. OrthogonalLine(A_{10}, a_9) → e_{13}
  110. Circle(A_{10}, B_6) → f_{10}
  111. Intersect(f_{10}, e_{13}, 2) → D_6
  112. Polygon(A_{10}, B_6, C_6, D_6) → poly1_3, g_4, h_6, i_4, j_4
  113. Segment(A_{11}, B_7) → a_{10}
  114. OrthogonalLine(B_7, a_{10}) → c_{14}
  115. Circle(B_7, A_{11}) → d_{14}
  116. Intersect(d_{14}, c_{14}, 2) → C_7
  117. OrthogonalLine(A_{11}, a_{10}) → e_{14}
  118. Circle(A_{11}, B_7) → f_{11}
  119. Intersect(f_{11}, e_{14}, 2) → D_7
  120. Polygon(A_{11}, B_7, C_7, D_7) → poly1_4, g_5, h_7, i_5, j_5
  121. Segment(A_{12}, B_8) → a_{11}
  122. OrthogonalLine(B_8, a_{11}) → c_{15}
  123. Circle(B_8, A_{12}) → d_{15}
  124. Intersect(d_{15}, c_{15}, 2) → C_8
  125. OrthogonalLine(A_{12}, a_{11}) → e_{15}
  126. Circle(A_{12}, B_8) → f_{12}
  127. Intersect(f_{12}, e_{15}, 2) → D_8
  128. Polygon(A_{12}, B_8, C_8, D_8) → poly1_5, g_6, h_8, i_6, j_6
  129. Segment(A_{13}, B_9) → a_{12}
  130. OrthogonalLine(B_9, a_{12}) → c_{16}
  131. Circle(B_9, A_{13}) → d_{16}
  132. Intersect(d_{16}, c_{16}, 2) → C_9
  133. OrthogonalLine(A_{13}, a_{12}) → e_{16}
  134. Circle(A_{13}, B_9) → f_{13}
  135. Intersect(f_{13}, e_{16}, 2) → D_9
  136. Polygon(A_{13}, B_9, C_9, D_9) → poly1_6, g_7, h_9, i_7, j_7
  137. Segment(A_{14}, B_{10}) → a_{13}
  138. OrthogonalLine(B_{10}, a_{13}) → c_{17}
  139. Circle(B_{10}, A_{14}) → d_{17}
  140. Intersect(d_{17}, c_{17}, 2) → C_{10}
  141. OrthogonalLine(A_{14}, a_{13}) → e_{17}
  142. Circle(A_{14}, B_{10}) → f_{14}
  143. Intersect(f_{14}, e_{17}, 2) → D_{10}
  144. Polygon(A_{14}, B_{10}, C_{10}, D_{10}) → poly1_7, g_8, h_{10}, i_8, j_8
  145. Segment(A_{15}, B_{11}) → a_{14}
  146. OrthogonalLine(B_{11}, a_{14}) → c_{18}
  147. Circle(B_{11}, A_{15}) → d_{18}
  148. Intersect(d_{18}, c_{18}, 2) → C_{11}
  149. OrthogonalLine(A_{15}, a_{14}) → e_{18}
  150. Circle(A_{15}, B_{11}) → f_{15}
  151. Intersect(f_{15}, e_{18}, 2) → D_{11}
  152. Polygon(A_{15}, B_{11}, C_{11}, D_{11}) → poly1_8, g_9, h_{11}, i_9, j_9
  153. Segment(A_{16}, B_{12}) → a_{15}
  154. OrthogonalLine(B_{12}, a_{15}) → c_{19}
  155. Circle(B_{12}, A_{16}) → d_{19}
  156. Intersect(d_{19}, c_{19}, 2) → C_{12}
  157. OrthogonalLine(A_{16}, a_{15}) → e_{19}
  158. Circle(A_{16}, B_{12}) → f_{16}
  159. Intersect(f_{16}, e_{19}, 2) → D_{12}
  160. Polygon(A_{16}, B_{12}, C_{12}, D_{12}) → poly1_9, g_{10}, h_{12}, i_{10}, j_{10}
  161. Segment(A_{17}, B_{13}) → a_{16}
  162. OrthogonalLine(B_{13}, a_{16}) → c_{20}
  163. Circle(B_{13}, A_{17}) → d_{20}
  164. Intersect(d_{20}, c_{20}, 2) → C_{13}
  165. OrthogonalLine(A_{17}, a_{16}) → e_{20}
  166. Circle(A_{17}, B_{13}) → f_{17}
  167. Intersect(f_{17}, e_{20}, 2) → D_{13}
  168. Polygon(A_{17}, B_{13}, C_{13}, D_{13}) → poly1_{10}, g_{11}, h_{13}, i_{11}, j_{11}
  169. Segment(A_{18}, B_{14}) → a_{17}
  170. OrthogonalLine(B_{14}, a_{17}) → c_{21}
  171. Circle(B_{14}, A_{18}) → d_{21}
  172. Intersect(d_{21}, c_{21}, 2) → C_{14}
  173. OrthogonalLine(A_{18}, a_{17}) → e_{21}
  174. Circle(A_{18}, B_{14}) → f_{18}
  175. Intersect(f_{18}, e_{21}, 2) → D_{14}
  176. Polygon(A_{18}, B_{14}, C_{14}, D_{14}) → poly1_{11}, g_{12}, h_{14}, i_{12}, j_{12}
  177. Segment(A_{19}, B_{15}) → a_{18}
  178. OrthogonalLine(B_{15}, a_{18}) → c_{22}
  179. Circle(B_{15}, A_{19}) → d_{22}
  180. Intersect(d_{22}, c_{22}, 2) → C_{15}
  181. OrthogonalLine(A_{19}, a_{18}) → e_{22}
  182. Circle(A_{19}, B_{15}) → f_{19}
  183. Intersect(f_{19}, e_{22}, 2) → D_{15}
  184. Polygon(A_{19}, B_{15}, C_{15}, D_{15}) → poly1_{12}, g_{13}, h_{15}, i_{13}, j_{13}
  185. Segment(A_{20}, B_{16}) → a_{19}
  186. OrthogonalLine(B_{16}, a_{19}) → c_{23}
  187. Circle(B_{16}, A_{20}) → d_{23}
  188. Intersect(d_{23}, c_{23}, 2) → C_{16}
  189. OrthogonalLine(A_{20}, a_{19}) → e_{23}
  190. Circle(A_{20}, B_{16}) → f_{20}
  191. Intersect(f_{20}, e_{23}, 2) → D_{16}
  192. Polygon(A_{20}, B_{16}, C_{16}, D_{16}) → poly1_{13}, g_{14}, h_{16}, i_{14}, j_{14}
  193. Segment(G_1, I_1) → b_5
  194. OrthogonalLine(I_1, b_5) → k_3
  195. Point(k_3) → J_1
  196. Line(J_1, b_5) → l_4
  197. OrthogonalLine(G_1, b_5) → m_4
  198. Intersect(l_4, m_4) → K_1
  199. Polygon(G_1, I_1, J_1, K_1) → poly2_1, g_{15}, i_{15}, j_{15}, k_2
  200. Segment(G_2, I_2) → b_6
  201. OrthogonalLine(I_2, b_6) → k_5
  202. Point(k_5) → J_2
  203. Line(J_2, b_6) → l_5
  204. OrthogonalLine(G_2, b_6) → m_5
  205. Intersect(l_5, m_5) → K_2
  206. Polygon(G_2, I_2, J_2, K_2) → poly2_2, g_{16}, i_{16}, j_{16}, k_4
  207. Segment(G_3, I_3) → b_7
  208. OrthogonalLine(I_3, b_7) → k_7
  209. Point(k_7) → J_3
  210. Line(J_3, b_7) → l_6
  211. OrthogonalLine(G_3, b_7) → m_6
  212. Intersect(l_6, m_6) → K_3
  213. Polygon(G_3, I_3, J_3, K_3) → poly2_3, g_{17}, i_{17}, j_{17}, k_6
  214. Segment(G_4, I_4) → b_8
  215. OrthogonalLine(I_4, b_8) → k_9
  216. Point(k_9) → J_4
  217. Line(J_4, b_8) → l_7
  218. OrthogonalLine(G_4, b_8) → m_7
  219. Intersect(l_7, m_7) → K_4
  220. Polygon(G_4, I_4, J_4, K_4) → poly2_4, g_{18}, i_{18}, j_{18}, k_8
  221. Segment(G_5, I_5) → b_9
  222. OrthogonalLine(I_5, b_9) → k_{11}
  223. Point(k_{11}) → J_5
  224. Line(J_5, b_9) → l_8
  225. OrthogonalLine(G_5, b_9) → m_8
  226. Intersect(l_8, m_8) → K_5
  227. Polygon(G_5, I_5, J_5, K_5) → poly2_5, g_{19}, i_{19}, j_{19}, k_{10}
  228. Segment(G_6, I_6) → b_{10}
  229. OrthogonalLine(I_6, b_{10}) → k_{13}
  230. Point(k_{13}) → J_6
  231. Line(J_6, b_{10}) → l_9
  232. OrthogonalLine(G_6, b_{10}) → m_9
  233. Intersect(l_9, m_9) → K_6
  234. Polygon(G_6, I_6, J_6, K_6) → poly2_6, g_{20}, i_{20}, j_{20}, k_{12}
  235. Segment(G_7, I_7) → b_{11}
  236. OrthogonalLine(I_7, b_{11}) → k_{15}
  237. Point(k_{15}) → J_7
  238. Line(J_7, b_{11}) → l_{10}
  239. OrthogonalLine(G_7, b_{11}) → m_{10}
  240. Intersect(l_{10}, m_{10}) → K_7
  241. Polygon(G_7, I_7, J_7, K_7) → poly2_7, g_{21}, i_{21}, j_{21}, k_{14}
  242. Segment(R_1, S_1) → s_3
  243. OrthogonalLine(S_1, s_3) → t_3
  244. Point(t_3) → T_1
  245. Line(T_1, s_3) → a_{20}
  246. OrthogonalLine(R_1, s_3) → b_{12}
  247. Intersect(b_{12}, a_{20}) → U_1
  248. Polygon(R_1, S_1, T_1, U_1) → poly3_1, r_2, s_2, t_2, u_1
  249. Segment(R_2, S_2) → s_5
  250. OrthogonalLine(S_2, s_5) → t_5
  251. Point(t_5) → T_2
  252. Line(T_2, s_5) → a_{21}
  253. OrthogonalLine(R_2, s_5) → b_{13}
  254. Intersect(b_{13}, a_{21}) → U_2
  255. Polygon(R_2, S_2, T_2, U_2) → poly3_2, r_3, s_4, t_4, u_2
  256. Segment(R_3, S_3) → s_7
  257. OrthogonalLine(S_3, s_7) → t_7
  258. Point(t_7) → T_3
  259. Line(T_3, s_7) → a_{22}
  260. OrthogonalLine(R_3, s_7) → b_{14}
  261. Intersect(b_{14}, a_{22}) → U_3
  262. Polygon(R_3, S_3, T_3, U_3) → poly3_3, r_4, s_6, t_6, u_3
  263. Segment(R_4, S_4) → s_9
  264. OrthogonalLine(S_4, s_9) → t_9
  265. Point(t_9) → T_4
  266. Line(T_4, s_9) → a_{23}
  267. OrthogonalLine(R_4, s_9) → b_{15}
  268. Intersect(b_{15}, a_{23}) → U_4
  269. Polygon(R_4, S_4, T_4, U_4) → poly3_4, r_5, s_8, t_8, u_4
  270. Segment(R_5, S_5) → s_{11}
  271. OrthogonalLine(S_5, s_{11}) → t_{11}
  272. Point(t_{11}) → T_5
  273. Line(T_5, s_{11}) → a_{24}
  274. OrthogonalLine(R_5, s_{11}) → b_{16}
  275. Intersect(b_{16}, a_{24}) → U_5
  276. Polygon(R_5, S_5, T_5, U_5) → poly3_5, r_6, s_{10}, t_{10}, u_5
  277. Segment(R_6, S_6) → s_{13}
  278. OrthogonalLine(S_6, s_{13}) → t_{13}
  279. Point(t_{13}) → T_6
  280. Line(T_6, s_{13}) → a_{25}
  281. OrthogonalLine(R_6, s_{13}) → b_{17}
  282. Intersect(b_{17}, a_{25}) → U_6
  283. Polygon(R_6, S_6, T_6, U_6) → poly3_6, r_7, s_{12}, t_{12}, u_6
  284. Segment(R_7, S_7) → s_{15}
  285. OrthogonalLine(S_7, s_{15}) → t_{15}
  286. Point(t_{15}) → T_7
  287. Line(T_7, s_{15}) → a_{26}
  288. OrthogonalLine(R_7, s_{15}) → b_{18}
  289. Intersect(b_{18}, a_{26}) → U_7
  290. Polygon(R_7, S_7, T_7, U_7) → poly3_7, r_8, s_{14}, t_{14}, u_7
  291. Segment(F_1, H_1) → p_1
  292. OrthogonalLine(H_1, p_1) → q_1
  293. Point(q_1) → L_1
  294. Point(q_1) → M_1
  295. Line(L_1, p_1) → p_2
  296. OrthogonalLine(F_1, p_1) → q_2
  297. Intersect(q_2, p_2) → N_1
  298. Polygon(F_1, H_1, L_1, N_1) → poly6, f_{21}, h_{17}, l_{11}, n_4
  299. Point(yAxis) → Z
  300. Segment(A_1, O_1) → n
  301. OrthogonalLine(O_1, n) → p
  302. Point(p) → P_1
  303. Line(P_1, n) → q
  304. OrthogonalLine(A_1, n) → r
  305. Intersect(q, r) → Q_1
  306. Polygon(A_1, O_1, P_1, Q_1) → poly4, a_2, o_1, p_3, q_3
  307. Segment(A_{21}, O_2) → n_5
  308. OrthogonalLine(O_2, n_5) → p_5
  309. Point(p_5) → P_2
  310. Line(P_2, n_5) → q_5
  311. OrthogonalLine(A_{21}, n_5) → r_9
  312. Intersect(q_5, r_9) → Q_2
  313. Polygon(A_{21}, O_2, P_2, Q_2) → poly4_5, a_{27}, o_2, p_4, q_4
  314. Segment(A_{22}, O_3) → n_6
  315. OrthogonalLine(O_3, n_6) → p_7
  316. Point(p_7) → P_3
  317. Line(P_3, n_6) → q_7
  318. OrthogonalLine(A_{22}, n_6) → r_{10}
  319. Intersect(q_7, r_{10}) → Q_3
  320. Polygon(A_{22}, O_3, P_3, Q_3) → poly4_6, a_{28}, o_3, p_6, q_6
  321. Segment(A_{23}, O_4) → n_7
  322. OrthogonalLine(O_4, n_7) → p_9
  323. Point(p_9) → P_4
  324. Line(P_4, n_7) → q_9
  325. OrthogonalLine(A_{23}, n_7) → r_{11}
  326. Intersect(q_9, r_{11}) → Q_4
  327. Polygon(A_{23}, O_4, P_4, Q_4) → poly4_7, a_{29}, o_4, p_8, q_8
  328. Segment(A_{24}, O_5) → n_8
  329. OrthogonalLine(O_5, n_8) → p_{11}
  330. Point(p_{11}) → P_5
  331. Line(P_5, n_8) → q_{11}
  332. OrthogonalLine(A_{24}, n_8) → r_{12}
  333. Intersect(q_{11}, r_{12}) → Q_5
  334. Polygon(A_{24}, O_5, P_5, Q_5) → poly4_8, a_{30}, o_5, p_{10}, q_{10}
  335. Segment(A_{25}, O_6) → n_9
  336. OrthogonalLine(O_6, n_9) → p_{13}
  337. Point(p_{13}) → P_6
  338. Line(P_6, n_9) → q_{13}
  339. OrthogonalLine(A_{25}, n_9) → r_{13}
  340. Intersect(q_{13}, r_{13}) → Q_6
  341. Polygon(A_{25}, O_6, P_6, Q_6) → poly4_9, a_{31}, o_6, p_{12}, q_{12}
  342. Segment(A_{26}, O_7) → n_{10}
  343. OrthogonalLine(O_7, n_{10}) → p_{15}
  344. Point(p_{15}) → P_7
  345. Line(P_7, n_{10}) → q_{15}
  346. OrthogonalLine(A_{26}, n_{10}) → r_{14}
  347. Intersect(q_{15}, r_{14}) → Q_7
  348. Polygon(A_{26}, O_7, P_7, Q_7) → poly4_{10}, a_{32}, o_7, p_{14}, q_{14}
  349. Segment(F_2, H_2) → d_1
  350. OrthogonalLine(H_2, d_1) → e_1
  351. Point(e_1) → L_2
  352. Point(e_1) → M_2
  353. OrthogonalLine(F_2, d_1) → m_{11}
  354. Line(L_2, d_1) → n_{11}
  355. Intersect(n_{11}, m_{11}) → N_2
  356. Polygon(F_2, H_2, L_2, N_2) → poly7, f_{22}, h_{18}, l_{12}, n_{12}
  357. Point(m_{11}) → F_3
  358. Segment(F_4, H_3) → d_{24}
  359. OrthogonalLine(H_3, d_{24}) → e_{24}
  360. Point(e_{24}) → L_3
  361. Point(e_{24}) → M_3
  362. OrthogonalLine(F_4, d_{24}) → m_{12}
  363. Line(L_3, d_{24}) → n_{14}
  364. Intersect(n_{14}, m_{12}) → N_3
  365. Polygon(F_4, H_3, L_3, N_3) → poly7_1, f_{23}, h_{19}, l_{13}, n_{13}
  366. Point(m_{12}) → F_5
  367. Segment(F_6, H_4) → d_{25}
  368. OrthogonalLine(H_4, d_{25}) → e_{25}
  369. Point(e_{25}) → L_4
  370. Point(e_{25}) → M_4
  371. OrthogonalLine(F_6, d_{25}) → m_{13}
  372. Line(L_4, d_{25}) → n_{16}
  373. Intersect(n_{16}, m_{13}) → N_4
  374. Polygon(F_6, H_4, L_4, N_4) → poly7_2, f_{24}, h_{20}, l_{14}, n_{15}
  375. Point(m_{13}) → F_7
  376. Segment(F_8, H_5) → d_{26}
  377. OrthogonalLine(H_5, d_{26}) → e_{26}
  378. Point(e_{26}) → L_5
  379. Point(e_{26}) → M_5
  380. OrthogonalLine(F_8, d_{26}) → m_{14}
  381. Line(L_5, d_{26}) → n_{18}
  382. Intersect(n_{18}, m_{14}) → N_5
  383. Polygon(F_8, H_5, L_5, N_5) → poly7_3, f_{25}, h_{21}, l_{15}, n_{17}
  384. Point(m_{14}) → F_9
  385. Segment(F_{10}, H_6) → d_{27}
  386. OrthogonalLine(H_6, d_{27}) → e_{27}
  387. Point(e_{27}) → L_6
  388. Point(e_{27}) → M_6
  389. OrthogonalLine(F_{10}, d_{27}) → m_{15}
  390. Line(L_6, d_{27}) → n_{20}
  391. Intersect(n_{20}, m_{15}) → N_6
  392. Polygon(F_{10}, H_6, L_6, N_6) → poly7_4, f_{26}, h_{22}, l_{16}, n_{19}
  393. Point(m_{15}) → F_{11}
  394. Segment(F_{12}, H_7) → d_{28}
  395. OrthogonalLine(H_7, d_{28}) → e_{28}
  396. Point(e_{28}) → L_7
  397. Point(e_{28}) → M_7
  398. OrthogonalLine(F_{12}, d_{28}) → m_{16}
  399. Line(L_7, d_{28}) → n_{22}
  400. Intersect(n_{22}, m_{16}) → N_7
  401. Polygon(F_{12}, H_7, L_7, N_7) → poly7_5, f_{27}, h_{23}, l_{17}, n_{21}
  402. Point(m_{16}) → F_{13}
  403. Segment(F_{14}, H_8) → d_{29}
  404. OrthogonalLine(H_8, d_{29}) → e_{29}
  405. Point(e_{29}) → L_8
  406. Point(e_{29}) → M_8
  407. OrthogonalLine(F_{14}, d_{29}) → m_{17}
  408. Line(L_8, d_{29}) → n_{24}
  409. Intersect(n_{24}, m_{17}) → N_8
  410. Polygon(F_{14}, H_8, L_8, N_8) → poly7_6, f_{28}, h_{24}, l_{18}, n_{23}
  411. Point(m_{17}) → F_{15}
  412. Segment(F_{16}, H_9) → d_{30}
  413. OrthogonalLine(H_9, d_{30}) → e_{30}
  414. Point(e_{30}) → L_9
  415. Point(e_{30}) → M_9
  416. OrthogonalLine(F_{16}, d_{30}) → m_{18}
  417. Line(L_9, d_{30}) → n_{26}
  418. Intersect(n_{26}, m_{18}) → N_9
  419. Polygon(F_{16}, H_9, L_9, N_9) → poly7_7, f_{29}, h_{25}, l_{19}, n_{25}
  420. Point(m_{18}) → F_{17}
  421. Segment(F_{18}, H_{10}) → d_{31}
  422. OrthogonalLine(H_{10}, d_{31}) → e_{31}
  423. Point(e_{31}) → L_{10}
  424. Point(e_{31}) → M_{10}
  425. OrthogonalLine(F_{18}, d_{31}) → m_{19}
  426. Line(L_{10}, d_{31}) → n_{28}
  427. Intersect(n_{28}, m_{19}) → N_{10}
  428. Polygon(F_{18}, H_{10}, L_{10}, N_{10}) → poly7_8, f_{30}, h_{26}, l_{20}, n_{27}
  429. Point(m_{19}) → F_{19}
  430. Segment(F_{20}, H_{11}) → d_{32}
  431. OrthogonalLine(H_{11}, d_{32}) → e_{32}
  432. Point(e_{32}) → L_{11}
  433. Point(e_{32}) → M_{11}
  434. OrthogonalLine(F_{20}, d_{32}) → m_{20}
  435. Line(L_{11}, d_{32}) → n_{30}
  436. Intersect(n_{30}, m_{20}) → N_{11}
  437. Polygon(F_{20}, H_{11}, L_{11}, N_{11}) → poly7_9, f_{31}, h_{27}, l_{21}, n_{29}
  438. Point(m_{20}) → F_{21}
  439. Segment(E_4, E_5) → r_{15}
  440. OrthogonalLine(E_5, r_{15}) → k_{16}
  441. Point(k_{16}) → E_6
  442. Point(k_{16}) → V_7
  443. Line(E_6, r_{15}) → p_{16}
  444. OrthogonalLine(E_4, r_{15}) → q_{16}
  445. Intersect(p_{16}, q_{16}) → E_7
  446. Polygon(E_4, E_5, E_6, E_7) → poly8, e_{33}, e_{34}, e_{35}, e_{36}
  447. Segment(E_8, E_9) → r_{16}
  448. OrthogonalLine(E_9, r_{16}) → k_{17}
  449. Point(k_{17}) → E_{10}
  450. Point(k_{17}) → V_8
  451. Line(E_{10}, r_{16}) → p_{17}
  452. OrthogonalLine(E_8, r_{16}) → q_{17}
  453. Intersect(p_{17}, q_{17}) → E_{11}
  454. Polygon(E_8, E_9, E_{10}, E_{11}) → poly8_1, e_{37}, e_{38}, e_{39}, e_{40}
  455. Segment(E_{12}, E_{13}) → r_{17}
  456. OrthogonalLine(E_{13}, r_{17}) → k_{18}
  457. Point(k_{18}) → E_{14}
  458. Point(k_{18}) → V_9
  459. Line(E_{14}, r_{17}) → p_{18}
  460. OrthogonalLine(E_{12}, r_{17}) → q_{18}
  461. Intersect(p_{18}, q_{18}) → E_{15}
  462. Polygon(E_{12}, E_{13}, E_{14}, E_{15}) → poly8_2, e_{41}, e_{42}, e_{43}, e_{44}
  463. Segment(E_{16}, E_{17}) → r_{18}
  464. OrthogonalLine(E_{17}, r_{18}) → k_{19}
  465. Point(k_{19}) → E_{18}
  466. Point(k_{19}) → V_{10}
  467. Line(E_{18}, r_{18}) → p_{19}
  468. OrthogonalLine(E_{16}, r_{18}) → q_{19}
  469. Intersect(p_{19}, q_{19}) → E_{19}
  470. Polygon(E_{16}, E_{17}, E_{18}, E_{19}) → poly8_3, e_{45}, e_{46}, e_{47}, e_{48}
  471. Segment(E_{20}, E_{21}) → r_{19}
  472. OrthogonalLine(E_{21}, r_{19}) → k_{20}
  473. Point(k_{20}) → E_{22}
  474. Point(k_{20}) → V_{11}
  475. Line(E_{22}, r_{19}) → p_{20}
  476. OrthogonalLine(E_{20}, r_{19}) → q_{20}
  477. Intersect(p_{20}, q_{20}) → E_{23}
  478. Polygon(E_{20}, E_{21}, E_{22}, E_{23}) → poly8_4, e_{49}, e_{50}, e_{51}, e_{52}
  479. Segment(E_{24}, E_{25}) → r_{20}
  480. OrthogonalLine(E_{25}, r_{20}) → k_{21}
  481. Point(k_{21}) → E_{26}
  482. Point(k_{21}) → V_{12}
  483. Line(E_{26}, r_{20}) → p_{21}
  484. OrthogonalLine(E_{24}, r_{20}) → q_{21}
  485. Intersect(p_{21}, q_{21}) → E_{27}
  486. Polygon(E_{24}, E_{25}, E_{26}, E_{27}) → poly8_5, e_{53}, e_{54}, e_{55}, e_{56}
  487. Segment(E_{28}, E_{29}) → r_{21}
  488. OrthogonalLine(E_{29}, r_{21}) → k_{22}
  489. Point(k_{22}) → E_{30}
  490. Point(k_{22}) → V_{13}
  491. Line(E_{30}, r_{21}) → p_{22}
  492. OrthogonalLine(E_{28}, r_{21}) → q_{22}
  493. Intersect(p_{22}, q_{22}) → E_{31}
  494. Polygon(E_{28}, E_{29}, E_{30}, E_{31}) → poly8_6, e_{57}, e_{58}, e_{59}, e_{60}
  495. Segment(E_{32}, E_{33}) → r_{22}
  496. OrthogonalLine(E_{33}, r_{22}) → k_{23}
  497. Point(k_{23}) → E_{34}
  498. Point(k_{23}) → V_{14}
  499. Line(E_{34}, r_{22}) → p_{23}
  500. OrthogonalLine(E_{32}, r_{22}) → q_{23}
  501. Intersect(p_{23}, q_{23}) → E_{35}
  502. Polygon(E_{32}, E_{33}, E_{34}, E_{35}) → poly8_7, e_{61}, e_{62}, e_{63}, e_{64}
  503. Segment(E_{36}, E_{37}) → r_{23}
  504. OrthogonalLine(E_{37}, r_{23}) → k_{24}
  505. Point(k_{24}) → E_{38}
  506. Point(k_{24}) → V_{15}
  507. Line(E_{38}, r_{23}) → p_{24}
  508. OrthogonalLine(E_{36}, r_{23}) → q_{24}
  509. Intersect(p_{24}, q_{24}) → E_{39}
  510. Polygon(E_{36}, E_{37}, E_{38}, E_{39}) → poly8_8, e_{65}, e_{66}, e_{67}, e_{68}
  511. Segment(E_{40}, E_{41}) → r_{24}
  512. OrthogonalLine(E_{41}, r_{24}) → k_{25}
  513. Point(k_{25}) → E_{42}
  514. Point(k_{25}) → V_{16}
  515. Line(E_{42}, r_{24}) → p_{25}
  516. OrthogonalLine(E_{40}, r_{24}) → q_{25}
  517. Intersect(p_{25}, q_{25}) → E_{43}
  518. Polygon(E_{40}, E_{41}, E_{42}, E_{43}) → poly8_9, e_{69}, e_{70}, e_{71}, e_{72}
  519. Segment(E_{44}, E_{45}) → r_{25}
  520. OrthogonalLine(E_{45}, r_{25}) → k_{26}
  521. Point(k_{26}) → E_{46}
  522. Point(k_{26}) → V_{17}
  523. Line(E_{46}, r_{25}) → p_{26}
  524. OrthogonalLine(E_{44}, r_{25}) → q_{26}
  525. Intersect(p_{26}, q_{26}) → E_{47}
  526. Polygon(E_{44}, E_{45}, E_{46}, E_{47}) → poly8_{10}, e_{73}, e_{74}, e_{75}, e_{76}
  527. Segment(E_{48}, E_{49}) → r_{26}
  528. OrthogonalLine(E_{49}, r_{26}) → k_{27}
  529. Point(k_{27}) → E_{50}
  530. Point(k_{27}) → V_{18}
  531. Line(E_{50}, r_{26}) → p_{27}
  532. OrthogonalLine(E_{48}, r_{26}) → q_{27}
  533. Intersect(p_{27}, q_{27}) → E_{51}
  534. Polygon(E_{48}, E_{49}, E_{50}, E_{51}) → poly8_{11}, e_{77}, e_{78}, e_{79}, e_{80}
  535. Segment(E_{52}, E_{53}) → r_{27}
  536. OrthogonalLine(E_{53}, r_{27}) → k_{28}
  537. Point(k_{28}) → E_{54}
  538. Point(k_{28}) → V_{19}
  539. Line(E_{54}, r_{27}) → p_{28}
  540. OrthogonalLine(E_{52}, r_{27}) → q_{28}
  541. Intersect(p_{28}, q_{28}) → E_{55}
  542. Polygon(E_{52}, E_{53}, E_{54}, E_{55}) → poly8_{12}, e_{81}, e_{82}, e_{83}, e_{84}
  543. Segment(F_{22}, H_{12}) → p_{29}
  544. OrthogonalLine(H_{12}, p_{29}) → q_{29}
  545. Point(q_{29}) → L_{12}
  546. Point(q_{29}) → M_{12}
  547. Line(L_{12}, p_{29}) → p_{30}
  548. OrthogonalLine(F_{22}, p_{29}) → q_{30}
  549. Intersect(q_{30}, p_{30}) → N_{12}
  550. Segment(F_{23}, H_{13}) → p_{31}
  551. OrthogonalLine(H_{13}, p_{31}) → q_{31}
  552. Point(q_{31}) → L_{13}
  553. Point(q_{31}) → M_{13}
  554. Line(L_{13}, p_{31}) → p_{32}
  555. OrthogonalLine(F_{23}, p_{31}) → q_{32}
  556. Intersect(q_{32}, p_{32}) → N_{13}
  557. Segment(F_{24}, H_{14}) → p_{33}
  558. OrthogonalLine(H_{14}, p_{33}) → q_{33}
  559. Point(q_{33}) → L_{14}
  560. Point(q_{33}) → M_{14}
  561. Line(L_{14}, p_{33}) → p_{34}
  562. OrthogonalLine(F_{24}, p_{33}) → q_{34}
  563. Intersect(q_{34}, p_{34}) → N_{14}
  564. Segment(F_{25}, H_{15}) → p_{35}
  565. OrthogonalLine(H_{15}, p_{35}) → q_{35}
  566. Point(q_{35}) → L_{15}
  567. Point(q_{35}) → M_{15}
  568. Line(L_{15}, p_{35}) → p_{36}
  569. OrthogonalLine(F_{25}, p_{35}) → q_{36}
  570. Intersect(q_{36}, p_{36}) → N_{15}
  571. Segment(F_{26}, H_{16}) → p_{37}
  572. OrthogonalLine(H_{16}, p_{37}) → q_{37}
  573. Point(q_{37}) → L_{16}
  574. Point(q_{37}) → M_{16}
  575. Line(L_{16}, p_{37}) → p_{38}
  576. OrthogonalLine(F_{26}, p_{37}) → q_{38}
  577. Intersect(q_{38}, p_{38}) → N_{16}
  578. Segment(F_{27}, H_{17}) → p_{39}
  579. OrthogonalLine(H_{17}, p_{39}) → q_{39}
  580. Point(q_{39}) → L_{17}
  581. Point(q_{39}) → M_{17}
  582. Line(L_{17}, p_{39}) → p_{40}
  583. OrthogonalLine(F_{27}, p_{39}) → q_{40}
  584. Intersect(q_{40}, p_{40}) → N_{17}
  585. Segment(F_{28}, H_{18}) → p_{41}
  586. OrthogonalLine(H_{18}, p_{41}) → q_{41}
  587. Point(q_{41}) → L_{18}
  588. Point(q_{41}) → M_{18}
  589. Line(L_{18}, p_{41}) → p_{42}
  590. OrthogonalLine(F_{28}, p_{41}) → q_{42}
  591. Intersect(q_{42}, p_{42}) → N_{18}
  592. Polygon(F_{28}, H_{18}, L_{18}, N_{18}) → poly6_1, f_{32}, h_{28}, l_{22}, n_{31}
  593. Segment(V_{20}, W_7) → c_{24}
  594. OrthogonalLine(W_7, c_{24}) → d_{33}
  595. Point(d_{33}) → Z_7
  596. Line(Z_7, c_{24}) → e_{85}
  597. OrthogonalLine(V_{20}, c_{24}) → f_{33}
  598. Intersect(e_{85}, f_{33}) → A_{27}
  599. Polygon(V_{20}, W_7, Z_7, A_{27}) → poly4_{11}, v_5, w_5, z_5, a_{33}
  600. Segment(B_{17}, C_{17}) → h_{29}
  601. OrthogonalLine(C_{17}, h_{29}) → l_{23}
  602. Point(l_{23}) → D_{17}
  603. Line(D_{17}, h_{29}) → m_{21}
  604. OrthogonalLine(B_{17}, h_{29}) → n_{32}
  605. Intersect(n_{32}, m_{21}) → E_{56}
  606. Polygon(B_{17}, C_{17}, D_{17}, E_{56}) → poly5_3, b_{19}, c_{25}, d_{34}, e_{86}
  607. Segment(E_{57}, E_{58}) → r_{28}
  608. OrthogonalLine(E_{58}, r_{28}) → k_{29}
  609. Point(k_{29}) → E_{59}
  610. Point(k_{29}) → V_{21}
  611. Line(E_{59}, r_{28}) → p_{43}
  612. OrthogonalLine(E_{57}, r_{28}) → q_{43}
  613. Intersect(p_{43}, q_{43}) → E_{60}
  614. Polygon(E_{57}, E_{58}, E_{59}, E_{60}) → poly8_{13}, e_{87}, e_{88}, e_{89}, e_{90}