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Playing with Circles and Segments (PcG)

Playing with Circles and Segments (PcG)
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2020-03-19 18:33
CircleAngleCoordinate GeometryTransformationGeometric Construction
This example demonstrates a playful construction involving circles and line segments, where repeated intersection, rotation, and distance measurement generate a chain of nested circles and points, producing an aesthetically pleasing geometric pattern. Core mathematical concepts include circle definitions, line-circle intersections, rotational transformations, distance and angle measurement, and dependency relationships among geometric objects. Construction highlights: using GeoGebra commands Circle, Line, Intersect, Rotate, Distance, Angle, and Segment, the initial circle c0, point P0, and segment s0 are built; then rotating P0 yields P1, which serves as the center for a new circle c1; intersecting with the segment produces further points, and this process iterates through B1 to B13, creating a recursive structure. Suitable for middle/high school geometry lessons, math labs, or artistic math explorations, helping students understand geometric transformations and dynamic construction logic.

GeoGebra Commands & Parameters

  1. Circle(P0, r0) → c0
  2. Line(P0, P0_1) → s0
  3. Intersect(c0, s0, 2) → P1
  4. Circle(P1, Pc) → cP1
  5. Intersect(cP1, s0, 2) → B1
  6. Circle(B1, Pc) → cB1
  7. Intersect(cB1, s0, 2) → B2
  8. Circle(P0, B1) → c1
  9. Circle(B2, Pc) → cB2
  10. Circle(P0, B2) → c2
  11. Intersect(cB2, s0, 2) → B3
  12. Circle(P0, B3) → c3
  13. Circle(B3, Pc) → cB3
  14. Intersect(cB3, s0, 2) → B4
  15. Circle(B4, Pc) → cB4
  16. Intersect(cB4, s0, 2) → B5
  17. Circle(P0, B5) → c5
  18. Circle(B5, Pc) → cB5
  19. Intersect(cB5, s0, 2) → B6
  20. Circle(B6, Pc) → cB6
  21. Intersect(cB6, s0, 2) → B7
  22. Circle(P0, B7) → c7
  23. Circle(B7, Pc) → cB7
  24. Intersect(cB7, s0, 2) → B8
  25. Circle(B8, Pc) → cB8
  26. Intersect(cB8, s0, 2) → B9
  27. Circle(P0, B9) → c9
  28. Circle(B9, Pc) → cB9
  29. Intersect(cB9, s0, 2) → B10
  30. Circle(B10, Pc) → cB10
  31. Intersect(cB10, s0, 2) → B11
  32. Circle(P0, B11) → c11
  33. Circle(B11, Pc) → cB11
  34. Intersect(cB11, s0, 2) → B12
  35. Circle(B12, Pc) → cB12
  36. Intersect(cB12, s0, 2) → B13
  37. Circle(P0, B13) → c13
  38. Circle(B13, Pc) → cB13
  39. Intersect(cB13, s0, 2) → B14
  40. Circle(B14, Pc) → cB14
  41. Intersect(cB14, s0, 2) → B15
  42. Circle(P0, B15) → c15
  43. Circle(B15, Pc) → cB15
  44. Intersect(cB15, s0, 2) → B16
  45. Circle(B16, Pc) → cB16
  46. Intersect(cB16, s0, 2) → B17
  47. Circle(P0, B17) → c17
  48. Circle(B17, Pc) → cB17
  49. Intersect(cB17, s0, 2) → B18
  50. Circle(B18, Pc) → cB18
  51. Intersect(cB18, s0, 2) → B19
  52. Circle(P0, B19) → c19
  53. Circle(B19, Pc) → cB19
  54. Intersect(cB19, s0, 2) → B20
  55. Circle(B20, Pc) → cB20
  56. Intersect(cB20, s0, 2) → B21
  57. Circle(P0, B21) → c21
  58. Circle(B21, Pc) → cB21
  59. Intersect(cB21, s0, 2) → B22
  60. Circle(B22, Pc) → cB22
  61. Intersect(cB22, s0, 2) → B23
  62. Circle(P0, B23) → c23
  63. Circle(B23, Pc) → cB23
  64. Intersect(cB23, s0, 2) → B24
  65. Circle(B24, Pc) → cB24
  66. Intersect(cB24, s0, 2) → B25
  67. Circle(P0, B25) → c25
  68. Circle(B25, Pc) → cB25
  69. Intersect(cB25, s0, 2) → B26
  70. Circle(B26, Pc) → cB26
  71. Intersect(cB26, s0, 2) → B27
  72. Circle(P0, B27) → c27
  73. Circle(B27, Pc) → cB27
  74. Intersect(cB27, s0, 2) → B28
  75. Circle(B28, Pc) → cB28
  76. Intersect(cB28, s0, 2) → B29
  77. Circle(P0, B29) → c29
  78. Circle(B29, Pc) → cB29
  79. Intersect(cB29, s0, 2) → B30
  80. Circle(B30, Pc) → cB30
  81. Intersect(cB30, s0, 2) → B31
  82. Circle(P0, B31) → c31
  83. Circle(B31, Pc) → cB31
  84. Intersect(cB31, s0, 2) → B32
  85. Circle(B32, Pc) → cB32
  86. Intersect(cB32, s0, 2) → B33
  87. Circle(P0, B33) → c33
  88. Circle(B33, Pc) → cB33
  89. Intersect(cB33, s0, 2) → B34
  90. Circle(B34, Pc) → cB34
  91. Intersect(cB34, s0, 2) → B35
  92. Circle(P0, B35) → c35
  93. Circle(P0, B34) → c34
  94. Circle(P0, B32) → c32
  95. Circle(P0, B30) → c30
  96. Circle(P0, B28) → c28
  97. Circle(P0, B26) → c26
  98. Circle(P0, B24) → c24
  99. Circle(P0, B22) → c22
  100. Circle(P0, B20) → c20
  101. Circle(P0, B18) → c18
  102. Circle(P0, B16) → c16
  103. Circle(P0, B14) → c14
  104. Circle(P0, B12) → c12
  105. Circle(P0, B10) → c10
  106. Circle(P0, B8) → c8
  107. Circle(P0, B6) → c6
  108. Circle(P0, B4) → c4
  109. Rotate(P0_1, α, P1) → P1_1
  110. Line(P1, P1_1) → rP1
  111. Line(B34, rP1) → rB34
  112. Intersect(c35, rB34, 2) → F34
  113. Distance(B34, F34) → D34
  114. Intersect(c1, rP1, 2) → W1
  115. Angle(B1, P0, W1) → β_2
  116. Line(B33, rP1) → rB33
  117. Intersect(c34, rB33, 2) → F33
  118. Distance(B33, F33) → D33
  119. Line(B32, rP1) → rB32
  120. Intersect(c33, rB32, 2) → F32
  121. Distance(B32, F32) → D32
  122. Line(B31, rP1) → rB31
  123. Intersect(c32, rB31, 2) → F31
  124. Distance(B31, F31) → D31
  125. Line(B30, rP1) → rB30
  126. Intersect(c31, rB30, 2) → F30
  127. Distance(B30, F30) → D30
  128. Line(B29, rP1) → rB29
  129. Intersect(c30, rB29, 2) → F29
  130. Distance(B29, F29) → D29
  131. Line(B28, rP1) → rB28
  132. Intersect(c29, rB28, 2) → F28
  133. Distance(B28, F28) → D28
  134. Line(B27, rP1) → rB27
  135. Intersect(c28, rB27, 2) → F27
  136. Distance(B27, F27) → D27
  137. Line(B26, rP1) → rB26
  138. Intersect(c27, rB26, 2) → F26
  139. Distance(B26, F26) → D26
  140. Line(B25, rP1) → rB25
  141. Intersect(c26, rB25, 2) → F25
  142. Distance(B25, F25) → D25
  143. Line(B24, rP1) → rB24
  144. Intersect(c25, rB24, 2) → F24
  145. Distance(B24, F24) → D24
  146. Line(B23, rP1) → rB23
  147. Intersect(c24, rB23, 2) → F23
  148. Distance(B23, F23) → D23
  149. Line(B22, rP1) → rB22
  150. Intersect(c23, rB22, 2) → F22
  151. Distance(B22, F22) → D22
  152. Line(B21, rP1) → rB21
  153. Intersect(c22, rB21, 2) → F21
  154. Distance(B21, F21) → D21
  155. Line(B20, rP1) → rB20
  156. Intersect(c21, rB20, 2) → F20
  157. Distance(B20, F20) → D20
  158. Line(B19, rP1) → rB19
  159. Intersect(c20, rB19, 2) → F19
  160. Distance(B19, F19) → D19
  161. Line(B18, rP1) → rB18
  162. Intersect(c19, rB18, 2) → F18
  163. Distance(B18, F18) → D18
  164. Line(B17, rP1) → rB17
  165. Intersect(c18, rB17, 2) → F17
  166. Distance(B17, F17) → D17
  167. Line(B16, rP1) → rB16
  168. Intersect(c17, rB16, 2) → F16
  169. Distance(B16, F16) → D16
  170. Line(B15, rP1) → rB15
  171. Intersect(c16, rB15, 2) → F15
  172. Distance(B15, F15) → D15
  173. Line(B14, rP1) → rB14
  174. Intersect(c15, rB14, 2) → F14
  175. Distance(B14, F14) → D14
  176. Line(B13, rP1) → rB13
  177. Intersect(c14, rB13, 2) → F13
  178. Distance(B13, F13) → D13
  179. Line(B12, rP1) → rB12
  180. Intersect(c13, rB12, 2) → F12
  181. Distance(B12, F12) → D12
  182. Line(B11, rP1) → rB11
  183. Intersect(c12, rB11, 2) → F11
  184. Distance(B11, F11) → D11
  185. Line(B10, rP1) → rB10
  186. Intersect(c11, rB10, 2) → F10
  187. Distance(B10, F10) → D10
  188. Line(B9, rP1) → rB9
  189. Intersect(c10, rB9, 2) → F9
  190. Distance(B9, F9) → D9
  191. Line(B8, rP1) → rB8
  192. Intersect(c9, rB8, 2) → F8
  193. Distance(B8, F8) → D8
  194. Line(B7, rP1) → rB7
  195. Intersect(c8, rB7, 2) → F7
  196. Distance(B7, F7) → D7
  197. Line(B6, rP1) → rB6
  198. Intersect(c7, rB6, 2) → F6
  199. Distance(B6, F6) → D6
  200. Line(B5, rP1) → rB5
  201. Intersect(c6, rB5, 2) → F5
  202. Distance(B5, F5) → D5
  203. Line(B4, rP1) → rB4
  204. Intersect(c5, rB4, 2) → F4
  205. Distance(B4, F4) → D4
  206. Line(B3, rP1) → rB3
  207. Intersect(c4, rB3, 2) → F3
  208. Distance(B3, F3) → D3
  209. Line(B2, rP1) → rB2
  210. Intersect(c3, rB2, 2) → F2
  211. Distance(B2, F2) → D2
  212. Line(B1, rP1) → rB1
  213. Intersect(c2, rB1, 2) → F1
  214. Distance(B1, F1) → D1
  215. Rotate(K1, β_2 + α, W1) → W2
  216. Angle(B2, P0, W2) → β_3
  217. Rotate(K2, β_3 + α, W2) → W3
  218. Angle(B3, P0, W3) → β_4
  219. Rotate(K3, β_4 + α, W3) → W4
  220. Angle(B4, P0, W4) → β_5
  221. Rotate(K4, β_5 + α, W4) → W5
  222. Angle(B5, P0, W5) → β_6
  223. Rotate(K5, β_6 + α, W5) → W6
  224. Angle(B6, P0, W6) → β_7
  225. Rotate(K6, β_7 + α, W6) → W7
  226. Angle(B7, P0, W7) → β_8
  227. Rotate(K7, β_8 + α, W7) → W8
  228. Angle(B8, P0, W8) → β_9
  229. Rotate(K8, β_9 + α, W8) → W9
  230. Angle(B9, P0, W9) → γ_0
  231. Rotate(K9, γ_0 + α, W9) → W10
  232. Angle(B10, P0, W10) → γ_1
  233. Rotate(K10, γ_1 + α, W10) → W11
  234. Angle(B11, P0, W11) → γ_2
  235. Rotate(K11, γ_2 + α, W11) → W12
  236. Angle(B12, P0, W12) → γ_3
  237. Rotate(K12, γ_3 + α, W12) → W13
  238. Angle(B13, P0, W13) → γ_4
  239. Rotate(K13, γ_4 + α, W13) → W14
  240. Angle(B14, P0, W14) → γ_5
  241. Rotate(K14, γ_5 + α, W14) → W15
  242. Angle(B15, P0, W15) → γ_6
  243. Rotate(K15, γ_6 + α, W15) → W16
  244. Angle(B16, P0, W16) → γ_7
  245. Rotate(K16, γ_7 + α, W16) → W17
  246. Angle(B17, P0, W17) → γ_8
  247. Rotate(K17, γ_8 + α, W17) → W18
  248. Angle(B18, P0, W18) → γ_9
  249. Rotate(K18, γ_9 + α, W18) → W19
  250. Angle(B19, P0, W19) → δ_0
  251. Rotate(K19, δ_0 + α, W19) → W20
  252. Angle(B20, P0, W20) → δ_1
  253. Rotate(K20, δ_1 + α, W20) → W21
  254. Angle(B20, P0, W21) → δ_2
  255. Rotate(K21, δ_2 + α, W21) → W22
  256. Angle(B22, P0, W22) → δ_3
  257. Rotate(K22, δ_3 + α, W22) → W23
  258. Angle(B23, P0, W23) → δ_4
  259. Rotate(K23, δ_4 + α, W23) → W24
  260. Angle(B24, P0, W24) → δ_5
  261. Rotate(K24, δ_5 + α, W24) → W25
  262. Angle(B25, P0, W25) → δ_6
  263. Rotate(K25, δ_6 + α, W25) → W26
  264. Angle(B26, P0, W26) → δ_7
  265. Rotate(K26, δ_7 + α, W26) → W27
  266. Angle(B27, P0, W27) → δ_8
  267. Rotate(K27, δ_8 + α, W27) → W28
  268. Angle(B28, P0, W28) → δ_9
  269. Rotate(K28, δ_9 + α, W28) → W29
  270. Angle(B29, P0, W29) → ε_0
  271. Rotate(K29, ε_0 + α, W29) → W30
  272. Angle(B30, P0, W30) → ε_1
  273. Rotate(K30, ε_1 + α, W30) → W31
  274. Angle(B31, P0, W31) → ε_2
  275. Rotate(K31, ε_2 + α, W31) → W32
  276. Angle(B32, P0, W32) → ε_3
  277. Rotate(K32, ε_3 + α, W32) → W33
  278. Angle(B33, P0, W33) → ε_4
  279. Rotate(K33, ε_4 + α, W33) → W34
  280. Angle(B34, P0, W34) → ε_5
  281. Rotate(K34, ε_5 + α, W34) → W35
  282. Circle(B35, Pc) → cB35
  283. Intersect(cB35, s0, 2) → B36
  284. Circle(P0, B36) → c36
  285. Line(B35, rP1) → rB35
  286. Intersect(c36, rB35, 2) → F35
  287. Segment(B2, F2) → l_5
  288. Segment(B3, F3) → m_5
  289. Segment(B4, F4) → n_5
  290. Segment(B5, F5) → a_5
  291. Segment(B6, F6) → b_5
  292. Segment(B7, F7) → f_6
  293. Segment(B8, F8) → g_6
  294. Segment(B9, F9) → h_6
  295. Segment(B10, F10) → i_6
  296. Segment(B11, F11) → j_6
  297. Segment(B12, F12) → k_6
  298. Segment(B13, F13) → l_6
  299. Segment(B14, F14) → m_6
  300. Segment(B15, F15) → n_6
  301. Segment(B16, F16) → p_6
  302. Segment(B17, F17) → q_6
  303. Segment(B18, F18) → r_6
  304. Segment(B19, F19) → s_6
  305. Segment(B20, F20) → t_6
  306. Segment(B21, F21) → a_6
  307. Segment(B22, F22) → b_6
  308. Segment(B23, F23) → f_7
  309. Segment(B24, F24) → g_7
  310. Segment(B25, F25) → h_7
  311. Segment(B26, F26) → i_7
  312. Segment(B27, F27) → j_7
  313. Segment(B28, F28) → k_7
  314. Segment(B29, F29) → l_7
  315. Segment(B30, F30) → m_7
  316. Segment(B31, F31) → n_7
  317. Segment(B32, F32) → p_7
  318. Segment(B33, F33) → q_7
  319. Segment(B34, F34) → r_7
  320. Segment(B35, F35) → s_7
  321. Segment(P0, F2) → h_9
  322. Segment(P0, F3) → i_9
  323. Segment(P0, F4) → j_9
  324. Segment(P0, F5) → k_9
  325. Segment(P0, F6) → l_9
  326. Segment(P0, F7) → m_9
  327. Segment(P0, F8) → n_9
  328. Segment(P0, F9) → p_9
  329. Segment(P0, F10) → q_9
  330. Segment(P0, F11) → r_9
  331. Segment(P0, F12) → s_9
  332. Segment(P0, F13) → t_9
  333. Segment(P0, F14) → a_9
  334. Segment(P0, F15) → t_{12}
  335. Segment(P0, F16) → a_{12}
  336. Segment(P0, F17) → b_{12}
  337. Segment(P0, F18) → f_{13}
  338. Segment(P0, F19) → g_{13}
  339. Segment(P0, F20) → h_{13}
  340. Segment(P0, F21) → i_{13}
  341. Segment(P0, F22) → j_{13}
  342. Segment(P0, F23) → k_{13}
  343. Segment(P0, F24) → l_{13}
  344. Segment(P0, F25) → m_{13}
  345. Segment(P0, F25) → n_{13}
  346. Segment(P0, F26) → p_{13}
  347. Segment(P0, F27) → q_{13}
  348. Segment(P0, F28) → r_{13}
  349. Segment(P0, F29) → s_{13}
  350. Segment(P0, F30) → t_{13}
  351. Segment(P0, F30) → a_{13}
  352. Segment(P0, F31) → b_{13}
  353. Segment(P0, F32) → c_{13}
  354. Segment(P0, F33) → d_{13}
  355. Segment(P0, F34) → e_{13}
  356. Segment(P0, F35) → f_{14}
  357. Segment(P0, W1) → f
  358. Segment(P0, F1) → g
  359. Segment(P1, W1) → h
  360. Segment(B1, F1) → i
  361. Distance(B35, F35) → D35
  362. Angle(B35, P0, W35) → ε_6
  363. Rotate(K35, ε_6 + α, W35) → W36
  364. Segment(W1, W2) → j
  365. Segment(W2, W3) → k
  366. Segment(W3, W4) → l
  367. Segment(W4, W5) → m
  368. Segment(W5, W6) → n
  369. Segment(W6, W7) → p
  370. Segment(W7, W8) → q
  371. Segment(W8, W9) → r
  372. Segment(W9, W10) → s
  373. Segment(W10, W11) → t
  374. Segment(W11, W12) → a
  375. Segment(W12, W13) → b
  376. Segment(W13, W14) → c
  377. Segment(W14, W15) → d
  378. Segment(W15, W16) → e
  379. Segment(W16, W17) → f_1
  380. Segment(W17, W18) → g_1
  381. Segment(W18, W19) → h_1
  382. Segment(W19, W20) → i_1
  383. Segment(W20, W21) → j_1
  384. Segment(W21, W22) → k_1
  385. Segment(W22, W23) → l_1
  386. Segment(W23, W24) → m_1
  387. Segment(W24, W25) → n_1
  388. Segment(W25, W26) → p_1
  389. Segment(W26, W27) → q_1
  390. Segment(W27, W28) → r_1
  391. Segment(W28, W29) → s_1
  392. Segment(W29, W30) → t_1
  393. Segment(W30, W31) → a_1
  394. Segment(W31, W32) → b_1
  395. Segment(W32, W33) → c_1
  396. Segment(W33, W34) → d_1
  397. Segment(W34, W35) → e_1
  398. Segment(W35, W36) → f_2
  399. Segment(P0, W2) → g_2
  400. Segment(P0, W3) → h_2
  401. Segment(P0, W4) → i_2
  402. Segment(P0, W5) → j_2
  403. Segment(P0, W6) → k_2
  404. Segment(P0, W7) → l_2
  405. Segment(P0, W8) → m_2
  406. Segment(P0, W9) → n_2
  407. Segment(P0, W10) → p_2
  408. Segment(P0, W11) → q_2
  409. Segment(P0, W12) → r_2
  410. Segment(P0, W13) → s_2
  411. Segment(P0, W14) → t_2
  412. Segment(P0, W15) → a_2
  413. Segment(P0, W16) → b_2
  414. Segment(P0, W17) → c_2
  415. Segment(P0, W18) → d_2
  416. Segment(P0, W19) → e_2
  417. Segment(P0, W20) → f_3
  418. Segment(P0, W21) → g_3
  419. Segment(P0, W22) → h_3
  420. Segment(P0, W23) → i_3
  421. Segment(P0, W24) → j_3
  422. Segment(P0, W25) → k_3
  423. Segment(P0, W26) → l_3
  424. Segment(P0, W27) → m_3
  425. Segment(P0, W28) → n_3
  426. Segment(P0, W29) → p_3
  427. Segment(P0, W30) → q_3
  428. Segment(P0, W31) → r_3
  429. Segment(P0, W32) → s_3
  430. Segment(P0, W33) → t_3
  431. Segment(P0, W34) → a_3
  432. Segment(P0, W35) → b_3
  433. Segment(P0, W36) → c_3

AI Drawing Prompt Reference

Draw a geometric pattern consisting of nested circles and line segments. Start with an initial circle, then iteratively generate new circles and points by rotating points, measuring distances and angles, and finding intersections with a line segment, creating a recursive and aesthetically pleasing structure that demonstrates circle definitions, line-circle intersections, rotational transformations, and distance/angle measurements.
The prompt above was compiled from the original drawing commands and is for reference only.