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Example 6b: Everyday Observations in Geometric Construction

Example 6b: Everyday Observations in Geometric Construction
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2023-03-22 02:16
CirclePolygonAngleTransformationGeometric Construction
In collections Realidades de GeoGebra
① This example demonstrates everyday geometric observations through a rich construction of points, lines, circles, and polygons, highlighting symmetry and reflection phenomena in an interactive GeoGebra environment. ② Core mathematical concepts include axial symmetry (reflection), midpoints, perpendicular lines, circles and arcs, polygons, and their interrelations; it also incorporates integers, inequalities, and step functions, showing cross-disciplinary integration. ③ Construction highlights: from the summary, commands such as OrthogonalLine, Mirror, Midpoint, Segment, Intersect, Circle, Point, Ray, If, Polar, CircumcircleArc, UnitVector, Line, UnitOrthogonalVector, and Angle are used, with objects like C, E_1, E_2, E_3, b, s, M, F_1, a4, G_1, a, f_2, I, a1, C_1, a2, a3, B, N_2, A1, p_1, G, G', g, B1, C1, D1, Q_2, A2, E1, F1, B2, C2, D2, E2, F2, B3, C3, A3, D3, forming a complex symmetric network. ④ Suitable for middle/high school geometry lessons, introductory geometric construction courses, and teacher demonstrations of symmetry and reflection, as well as student exploration of transformations and construction logic.

GeoGebra Commands & Parameters

  1. OrthogonalLine(E_2, xAxis) → b
  2. OrthogonalLine(E_3, b) → s
  3. Mirror(E_1, s) → M
  4. Midpoint(E_1, M) → F_1
  5. Segment(F_1, E_1) → a4
  6. Mirror(F_1, M) → G_1
  7. Segment(G_1, E_1) → a
  8. OrthogonalLine(E_1, a) → f_2
  9. Intersect(f_2, b) → I
  10. Segment(E_1, I) → a1
  11. Intersect(b, s) → C_1
  12. Segment(I, C_1) → a2
  13. Segment(F_1, C_1) → a3
  14. Circle(B, 3.8) → p_1
  15. Point(p_1) → G
  16. Mirror(G, B) → G'
  17. Ray(B, G') → g
  18. RemoveUndefined(B1) → C1
  19. If(A1 ≟ Element[C1, 1], Element[C1, 2], Element[C1, 1]) → D1
  20. If(D1 ≟ Element[B1, 1], a1, If[D1 ≟ Element[B1, 2], a2, If[D1 ≟ Element[B1, 3], a3, a4]]) → E1
  21. Ray(D1, Vector[D1, Mirror[A1, OrthogonalLine[D1, E1]]]) → F1
  22. RemoveUndefined(B2) → C2
  23. If(A2 ≟ Element[C2, 1], Element[C2, 2], Element[C2, 1]) → D2
  24. If(D2 ≟ Element[B2, 1], a1, If[D2 ≟ Element[B2, 2], a2, If[D2 ≟ Element[B2, 3], a3, a4]]) → E2
  25. Ray(D2, Vector[D2, Mirror[A2, OrthogonalLine[D2, E2]]]) → F2
  26. RemoveUndefined(B3) → C3
  27. If(A3 ≟ Element[C3, 1], Element[C3, 2], Element[C3, 1]) → D3
  28. If(D3 ≟ Element[B3, 1], a1, If[D3 ≟ Element[B3, 2], a2, If[D3 ≟ Element[B3, 3], a3, a4]]) → E3
  29. Ray(D3, Vector[D3, Mirror[A3, OrthogonalLine[D3, E3]]]) → F3
  30. RemoveUndefined(B4) → C4
  31. If(A4 ≟ Element[C4, 1], Element[C4, 2], Element[C4, 1]) → D4
  32. Segment(D2, D3) → d3
  33. Segment(D1, D2) → d2
  34. Segment(A1, D1) → d1
  35. Segment(D3, D) → Q5
  36. Segment(D2, D) → Q5c
  37. Segment(D1, D) → Q5b
  38. Segment(B, D) → Q5a
  39. Circle(D, 0.12) → Q1
  40. Intersect(Q1, Q5, 1) → Q6
  41. Polar(D3, Q1) → Q2
  42. Intersect(Q1, Q2) → Q4, Q3
  43. CircumcircleArc(Q3, Q6, Q4) → Q7
  44. Intersect(Q7, Segment[D3, D4]) → Q8,
  45. Intersect(Q1, Q5c, 1) → Q6c
  46. Polar(D2, Q1) → Q2c
  47. Intersect(Q1, Q2c) → Q3c, Q4c
  48. CircumcircleArc(Q3c, Q6c, Q4c) → Q7c
  49. Intersect(Q7c, Segment[D2, D3]) → Q8c,
  50. Intersect(Q1, Q5b, 1) → Q6b
  51. Polar(D1, Q1) → Q2b
  52. Intersect(Q1, Q2b) → Q3b, Q4b
  53. CircumcircleArc(Q3b, Q6b, Q4b) → Q7b
  54. Intersect(Q7b, Segment[D1, D2]) → Q8b,
  55. Intersect(Q1, Q5a, 1) → Q6a
  56. Polar(B, Q1) → Q2a
  57. Intersect(Q1, Q2a) → Q3a, Q4a
  58. CircumcircleArc(Q3a, Q6a, Q4a) → Q7a
  59. Intersect(Q7a, Segment[B, D1]) → Q8a,
  60. If(Defined[Q8a], true, false) → choque0
  61. If(Defined[Q8b] ∧ (¬choque0), true, false) → choque1
  62. If(Defined[Q8c] ∧ (¬choque0) ∧ (¬choque1), true, false) → choque2
  63. If(Defined[Q8] ∧ (¬choque0) ∧ (¬choque1) ∧ (¬choque2), true, false) → choque3
  64. If(choque3, Q8, D4) → DF
  65. Ray(DF, D) → F6
  66. RemoveUndefined(B8) → C8
  67. If(A8 ≟ Element[C8, 1], Element[C8, 2], Element[C8, 1]) → D8
  68. UnitVector(Vector[D, D8]) → G6
  69. Segment(D, D8) → d6
  70. Segment(D3, DF) → d4
  71. If(choque2, Q8c, D3) → DFc
  72. Ray(DFc, D) → F15
  73. RemoveUndefined(B17) → C17
  74. If(A17 ≟ Element[C17, 1], Element[C17, 2], Element[C17, 1]) → D17
  75. Segment(D, D17) → d6c
  76. UnitVector(Vector[DFc, D17]) → G15
  77. Segment(D2, DFc) → d4c
  78. If(choque1, Q8b, D2) → DFb
  79. Ray(DFb, D) → F12
  80. RemoveUndefined(B14) → C14
  81. If(A14 ≟ Element[C14, 1], Element[C14, 2], Element[C14, 1]) → D14
  82. UnitVector(Vector[DFb, D14]) → G12
  83. Segment(D, D14) → d6b
  84. Segment(D1, DFb) → d4b
  85. If(choque0, Q8a, D1) → DFa
  86. Ray(DFa, D) → F9
  87. RemoveUndefined(B11) → C11
  88. If(A11 ≟ Element[C11, 1], Element[C11, 2], Element[C11, 1]) → D11
  89. UnitVector(Vector[D, D11]) → G9
  90. Segment(D, D11) → d6a
  91. Segment(B, DFa) → d4a
  92. Segment(A1, PP) → h
  93. UnitVector(Vector[A1, D1]) → G1
  94. Ray(B, DFa) → ja
  95. Mirror(D, ja) → Da'
  96. Mirror(DFa, ja) → DFFa
  97. Ray(DFFa, Da') → F8
  98. RemoveUndefined(B10) → C10
  99. If(A10 ≟ Element[C10, 1], Element[C10, 2], Element[C10, 1]) → D10
  100. Segment(DFa, D10) → d5a
  101. UnitVector(Vector[DFa, D10]) → G8
  102. UnitVector(Vector[D1, D2]) → G2
  103. Ray(D1, DFb) → jb
  104. Mirror(DFb, jb) → DFFb
  105. Mirror(D, jb) → Db'
  106. Ray(DFFb, Db') → F11
  107. RemoveUndefined(B13) → C13
  108. If(A13 ≟ Element[C13, 1], Element[C13, 2], Element[C13, 1]) → D13
  109. UnitVector(Vector[DFb, D13]) → G11
  110. Segment(DFb, D13) → d5b
  111. UnitVector(Vector[D2, D3]) → G3
  112. Ray(D2, DFc) → jc
  113. Mirror(D, jc) → Dc'
  114. Mirror(DFc, jc) → DFFc
  115. Ray(DFFc, Dc') → F14
  116. RemoveUndefined(B16) → C16
  117. If(A16 ≟ Element[C16, 1], Element[C16, 2], Element[C16, 1]) → D16
  118. Segment(DFc, D16) → d5c
  119. UnitVector(Vector[DFc, D16]) → G14
  120. Ray(D3, DF) → j
  121. Mirror(D, j) → D'
  122. Mirror(DF, j) → DFF
  123. Ray(DFF, D') → F5
  124. RemoveUndefined(B7) → C7
  125. If(A7 ≟ Element[C7, 1], Element[C7, 2], Element[C7, 1]) → D7
  126. Segment(A7, D7) → d5
  127. UnitVector(Vector[DF, D7]) → G5
  128. UnitVector(Vector[D3, DF]) → G4
  129. Intersect(s, yAxis) → E
  130. OrthogonalLine(E, s) → p
  131. OrthogonalLine(M, p) → r
  132. Intersect(s, g) → L
  133. OrthogonalLine(L, s) → k
  134. Mirror(B, k) → B'
  135. If(choque2, If[h - d1 - d2 > d4c, If[h - d4c - d1 - d2 < d6c, D + ((h - d4c - d1 - d2) * G15), D + (d6c * G15)], D], If[choque1, If[h - d1 > d4b, If[h - d4b - d1 < d6b, D + ((h - d4b - d1) * G12), D + (d6b * G12)], D], If[choque0, If[h > d4a, If[h - d4a < d6a, D + ((h - d4a) * G9), D + (d6a * G9)], D], If[choque3 ∧ h ≥ sd, If[h - sd < d6, D + ((h - sd) * G6), D + (d6 * G6)], D]]]) → DD
  136. Mirror(DD, s) → D'_1
  137. Mirror(D'_1, yAxis) → H
  138. Mirror(H, r) → O
  139. Line(D, s) → e
  140. Mirror(O, e) → U
  141. Line(U, e) → i_1
  142. Mirror(B, s) → N
  143. Mirror(N, yAxis) → G_2
  144. Intersect(g, e) → F
  145. Mirror(F, k) → F'
  146. Line(O, r) → b_1
  147. Mirror(F', s) → F''
  148. Circle(B, d) → c
  149. Mirror(mesa, s) → mesa2b
  150. Mirror(mesa2b, yAxis) → mesa3a
  151. Mirror(O, p) → Y
  152. Mirror(mesa, p) → mesa2a
  153. Mirror(mesa2b, r) → mesa3b
  154. Mirror(mesa3b, p) → mesa4
  155. Intersect(r, b) → A_1
  156. Mirror(A_1, M) → B_1
  157. Segment(B_1, A_1) → j_1
  158. Mirror(C_1, E) → D_1
  159. Segment(D_1, C_1) → k_1
  160. Segment(B, G) → l_1
  161. Intersect(c, l_1, 1) → A
  162. UnitOrthogonalVector(v_1) → w
  163. If(D4 ≟ Element[B4, 1], a1, If[D4 ≟ Element[B4, 2], a2, If[D4 ≟ Element[B4, 3], a3, a4]]) → E4
  164. Ray(D4, Vector[D4, Mirror[A4, OrthogonalLine[D4, E4]]]) → F4
  165. UnitVector(Vector[D3, D]) → u
  166. If(choque2, If[h < d1, A1 + (h * G1), If[h - d1 < d2, D1 + ((h - d1) * G2), If[h - d1 - d2 < d4c, D2 + ((h - d1 - d2) * G3), If[h - d1 - d2 - d4c < d5c, DFc + ((h - d4c - d1 - d2) * G14), DFc + (d5c * G14)]]]], If[choque1, If[h < d1, A1 + (h * G1), If[h - d1 < d4b, D1 + ((h - d1) * G2), If[h - d1 - d4b < d5b, DFb + ((h - d4b - d1) * G11), DFb + (d5b * G11)]]], If[choque0, If[h < d4a, A1 + (h * G1), If[h < d4a + d5a, DFa + ((h - d4a) * G8), DFa + (d5a * G8)]], If[h < d1, A1 + (h * G1), If[h - d1 < d2, D1 + ((h - d1) * G2), If[h - d1 - d2 < d3, D2 + ((h - d1 - d2) * G3), If[h - sd < 0, D3 + ((h - d1 - d2 - d3) * G4), If[choque3, If[h - sd < d5, DF + ((h - sd) * G5), DF + (d5 * G5)], D3 + (d4 * G4)]]]]]]]) → P
  167. Segment(DF, P) → r5
  168. Segment(DF, DD) → r5'
  169. Mirror(P, k_1) → P'_2
  170. Mirror(P'_2, j_1) → P''_1
  171. Mirror(P''_1, a) → P'''
  172. Mirror(P'_2, a) → P''
  173. Mirror(P, a) → P'
  174. Mirror(DF, k_1) → DF'
  175. Mirror(DF', j_1) → DF''
  176. Mirror(DF'', a) → DF'''
  177. Intersect(E1, OrthogonalLine[D2, E1]) → H1
  178. Intersect(E1, OrthogonalLine[A1, E1]) → I1
  179. Intersect(E2, OrthogonalLine[D3, E2]) → H2
  180. Intersect(E2, OrthogonalLine[A2, E2]) → I2
  181. Intersect(E3, OrthogonalLine[D4, E3]) → H3
  182. Intersect(E3, OrthogonalLine[A3, E3]) → I3
  183. Angle(H1, D1, D2) → J1
  184. Angle(A1, D1, I1) → K1
  185. Angle(H2, D2, D3) → J2
  186. Angle(H3, D3, D4) → J3
  187. Angle(A2, D2, I2) → K2
  188. Angle(A3, D3, I3) → K3
  189. Segment(B, P) → r1
  190. Segment(B, D1) → r1a
  191. Segment(D1, P) → r2
  192. Segment(D2, D1) → r2a
  193. Segment(D2, P) → r3
  194. Segment(D2, D3) → r3a
  195. Segment(D3, P) → r4
  196. Segment(D3, DF) → r4a
  197. Mirror(D1, p) → D1m1a
  198. Mirror(D1, s) → D1m1b
  199. Mirror(D2, p) → D2m1a
  200. Mirror(D2, s) → D2m1b
  201. Segment(D1m1b, D2m1b) → m
  202. Mirror(D3, E) → D3m2a
  203. Mirror(D3m2a, M) → D3m2b
  204. Mirror(D3m2a, r) → D3m3
  205. Mirror(D4, E) → D4m2a
  206. Mirror(D4m2a, r) → D4m3
  207. Mirror(DF, E) → DFm2a
  208. Mirror(DFm2a, r) → DFm3
  209. Segment(D3m3, DFm3) → i
  210. Segment(D1m1a, D2m1a) → q
  211. Mirror(D2m1a, s) → D2m2a
  212. Segment(D2m2a, D3m2a) → l
  213. Mirror(D2m1b, r) → D2m2b
  214. Segment(D2m2b, D3m2b) → a_1
  215. Mirror(B, p) → Bm1a
  216. Mirror(B, s) → Bm1b
  217. Mirror(B, E) → Bm2a
  218. Mirror(Bm2a, M) → Bm2b
  219. Mirror(Bm2a, r) → Bm3
  220. Mirror(DD, p) → Z
  221. Segment(B, DFa) → r1aa
  222. Segment(B, P) → r1A
  223. Segment(P, DFa) → r1Ab
  224. Segment(DFa, DD) → r5A
  225. Segment(D1, P) → r1B
  226. Segment(D1, DFb) → n
  227. Segment(P, DFb) → r1Bb
  228. Segment(DFb, DD) → r5B
  229. Segment(D2, P) → c_1
  230. Segment(D2, DFc) → d_1
  231. Segment(DFc, P) → e_1
  232. Segment(D16, DFc) → f_1
  233. Segment(DD, DFc) → r5C
  234. Segment(B, DFa) → m_1
  235. If(choque3, Q8, If[choque0, Q8a, If[choque1, Q8b, If[choque2, Q8c, D4]]]) → DT
  236. Mirror(DT, p) → DTm1a
  237. Segment(D1m1a, DTm1a) → n_1
  238. Mirror(DT, s) → DTm1b
  239. Segment(D1m1b, DTm1b) → q_1
  240. Mirror(DTm1a, s) → DTm2a
  241. Segment(D2m2a, DTm2a) → r_1
  242. Mirror(DTm1b, r) → DTm2b
  243. Segment(D2m2b, DTm2b) → s_1
  244. If(D1 ≟ Element[B1, 2] ∨ D1 ≟ Element[B1, 4], 1, 2) → test1
  245. If((D2 ≟ Element[B2, 1] ∨ D2 ≟ Element[B2, 3]) ∧ test1 ≟ 1 ∨ (D2 ≟ Element[B2, 2] ∨ D2 ≟ Element[B2, 4]) ∧ test1 ≟ 2, 1, 2) → test2
  246. Segment(E_1, E_3) → g_1
  247. Segment(E_3, C_1) → h_1
  248. Mirror(K, C) → K'
  249. Mirror(Q_1, C) → C_2
  250. Mirror(W_1, C) → D_2
  251. Mirror(N_1, C) → F_2
  252. Mirror(M_1, C) → H_2
  253. Mirror(L_1, C) → I_2
  254. Mirror(Q, C) → Q'
  255. Mirror(R, C) → R'
  256. Mirror(P_1, C) → J_2
  257. Mirror(O_1, C) → K_2
  258. Mirror(R_1, C) → L_2
  259. Mirror(Z_1, C) → M_2
  260. Segment(C, K) → t_1
  261. Segment(K, Q) → a_2
  262. Segment(Q, J) → b_2
  263. Segment(J, C) → c_2
  264. Ray(N_2, Q_2) → e_2
  265. Circle(C + (0.5, 0.25), 3.5) → d_2
  266. Intersect(d_2, e_2, 1) → P_2
  267. Segment(N_2, P_2) → g_2
  268. Circle(C + (0.5, 0.25), 3.5) → h_2
  269. Line(R', I_2) → i_2
  270. Line(K_1, H_1) → j_2
  271. Mirror(O_2, j_2) → W_2
  272. Mirror(W_2, i_2) → Z_2
  273. Line(L_2, K_2) → k_2
  274. Mirror(Z_2, k_2) → A_3
  275. Line(Z_1, U_1) → l_2
  276. Mirror(S_2, l_2) → B_3
  277. Mirror(V_2, j_2) → C_3
  278. Mirror(C_3, i_2) → D_3
  279. Line(B_2, M_2) → m_2
  280. Mirror(S_2, m_2) → F_3
  281. Mirror(T_2, m_2) → G_3
  282. Mirror(Z_2, m_2) → H_3
  283. Mirror(H_3, i_2) → I_3
  284. Mirror(I_3, j_2) → J_3
  285. Line(U_1, K_2) → n_2
  286. Line(T_1, I_2) → p_2
  287. Line(Z_1, L_2) → q_2
  288. Line(O_1, R') → r_2
  289. Mirror(I_3, n_2) → K_3
  290. Mirror(J_3, n_2) → L_3
  291. Mirror(G_3, n_2) → M_3
  292. Mirror(L_3, p_2) → N_3
  293. Mirror(U_2, q_2) → O_3
  294. Mirror(V_2, q_2) → P_3
  295. Mirror(C_3, q_2) → Q_3
  296. Mirror(P_3, r_2) → R_3
  297. Segment(J, I_1) → s_2
  298. Segment(I_1, Z_1) → t_2
  299. Segment(Z_1, V_1) → a_3
  300. Segment(V_1, J) → b_3

AI Drawing Prompt Reference

In a coordinate plane, start with a horizontal reference line and define a perpendicular line through a given point, forming a small network of intersecting lines. Choose another point on one line and construct its reflection across a designated line to obtain symmetric points and connect them with segments, then create further intersections between the generated lines and segments. Draw a circle centered at a selected point with radius about 3.8, and use polar/related constructions from the circle to generate additional intersection points. Finally, form one or more circumcircle arcs passing through the relevant intersection points and connect/trim them by the corresponding segments, producing a symmetric reflection pattern.
The prompt above was compiled from the original drawing commands and is for reference only.