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Superimposable Right Triangles Construction

Superimposable Right Triangles Construction
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2017-07-16 00:32
TriangleCircleAngleTransformationGeometric Construction
① This example demonstrates a composite figure of multiple right triangles that can be superimposed through translations, rotations, and reflections, highlighting the concept of congruent (superimposable) triangles and their positional changes on the plane. ② Core mathematical concepts include triangle congruence, geometric transformations (translation, rotation, reflection), auxiliary constructions with circles and arcs, and the use of piecewise functions and parametric curves to describe boundaries. ③ Construction highlights: from the summary, GeoGebra commands such as Semicircle, Point, Polygon, Segment, Circle, Intersect, Line, OrthogonalLine, Angle, CircleArc, Rotate, Ray, Vector, Translate, CircleSector, Midpoint, and Mirror are used. Objects include multiple triangles, circles, segments, and auxiliary points, with transformations generating the overlapping right triangles. ④ Suitable for middle school geometry or introductory college transformation geometry, useful for visualizing triangle congruence criteria and transformation properties, and as a dynamic geometry exploration example.

GeoGebra Commands & Parameters

  1. Semicircle(O_3, R_3) → s_6
  2. Point(s_6) → S_3
  3. Polygon(O_3, S_3, R_3) → poly11, r_7, o_3, s_7
  4. Point(Segment[Pos_D, W_1], v - 7) → D_9
  5. Segment(W_1, D_2) → l_5
  6. Segment(B_1, C_1) → a_2
  7. Circle(E_1, a_2) → e_3
  8. Point(e_3) → G_1
  9. Semicircle(E_1, G_1) → h_3
  10. Semicircle(B_1, C_1) → t_2
  11. Point(t_2) → D_1
  12. Segment(D_1, C_1) → f_3
  13. Circle(G_1, f_3) → k_3
  14. Intersect(h_3, k_3, 1) → H_1
  15. Segment(D_1, H_1) → a_3
  16. Circle(D, 6) → c_1
  17. Line(C, D) → s
  18. Intersect(c_1, s, 1) → F
  19. Circle(D, 7) → t_1
  20. Point(Segment[C, D], n) → N
  21. Segment(C, N) → f_1
  22. OrthogonalLine(D, f_1) → s_1
  23. Intersect(t_1, s_1, 1) → R
  24. Line(L, J) → g
  25. Segment(L, J) → h
  26. Circle(L, (m * h)) → d
  27. Intersect(d, g, 2) → G
  28. Segment(L, G) → i
  29. OrthogonalLine(J, h) → j
  30. Point(j) → H
  31. Segment(J, H) → l
  32. Circle(J, ((m - 1) * l)) → e
  33. Intersect(e, j) → I, K
  34. Segment(J, I) → k
  35. Angle(L, J, H) → γ
  36. Circle(J, 6) → t
  37. Circle(J, 4) → h_1
  38. Intersect(t, h, 1) → A
  39. Circle(J, 4) → c
  40. Intersect(c, l, 1) → B
  41. Point(Segment[J, A], m - 2) → M
  42. Segment(J, M) → f
  43. Point(Segment[J, B], m - 3) → M_1
  44. Segment(J, M_1) → q
  45. Point(Segment[A, B], m - 4) → M_2
  46. Segment(A, M_2) → r
  47. Polygon(A, J, B) → poly1, b, a, j_1
  48. OrthogonalLine(D, f_1) → g_1
  49. Point(Segment[D, R], n - 1) → N_1
  50. Segment(D, N_1) → i_1
  51. Intersect(c_1, f_1, 1) → F_1
  52. Point(Segment[D, F_1], n - 2) → N_2
  53. Segment(D, N_2) → k_1
  54. Circle(F, 7.5) → d_1
  55. Point(d_1) → O
  56. Intersect(d_1, i_1, 1) → P
  57. Circle(P, O) → e_1
  58. Intersect(d_1, e_1, 2) → Q
  59. Point(Arc[d_1, O, Q], n - 4) → N_4
  60. CircleArc(F_1, O, N_4) → p_1
  61. Point(Segment[F_1, O], n - 3) → N_3
  62. Segment(F_1, N_3) → l_1
  63. Segment(F_1, N_4) → m_1
  64. Point(Segment[F_1, P], n - 5) → N_5
  65. Segment(F_1, N_5) → n_1
  66. Point(Segment[D, P], n - 6) → N_6
  67. Segment(D, N_6) → q_1
  68. Polygon(F_1, D, P) → poly2, p_2, f_2, d_2
  69. Segment(F_1, P) → r_1
  70. Angle(F_1, D, R) → α
  71. OrthogonalLine(D, s) → a_1
  72. Line(E, S) → b_1
  73. Rotate(S, 50°, E) → S'
  74. Angle(S, E, S') → β
  75. Line(E, S') → g_2
  76. Circle(E, 7) → c_2
  77. Intersect(c_2, b_1, 2) → T
  78. OrthogonalLine(T, g_2) → h_2
  79. Intersect(g_2, h_2) → U
  80. Point(Segment[E, T], p) → P_1
  81. Segment(E, P_1) → j_2
  82. Ray(E, T) → k_2
  83. Point(k_2) → W
  84. Segment(E, W) → l_2
  85. Circle(E, W) → e_2
  86. Intersect(e_2, g_2, 2) → Z
  87. Point(Arc[e_2, W, Z], p - 1) → P_2
  88. Segment(E, P_2) → m_2
  89. Angle(W, E, P_2) → δ
  90. Segment(E, W) → i_2
  91. Point(h_2) → A_1
  92. Segment(T, A_1) → n_2
  93. Angle(E, U, T) → ε
  94. Point(Segment[T, U], p - 3) → P_3
  95. Segment(T, P_3) → q_2
  96. Point(Segment[E, U], p - 4) → P_4
  97. Segment(E, P_4) → r_2
  98. Point(Segment[T, A_1], p - 2) → P_{21}
  99. Segment(T, P_{21}) → s_2
  100. Angle(E, U, T) → ζ
  101. Segment(B_1, D_1) → b_2
  102. Polygon(B_1, D_1, C_1) → poly3, c_3, b_3, d_3
  103. Segment(E_1, G_1) → g_3
  104. Segment(E_1, H_1) → i_3
  105. Segment(H_1, G_1) → j_3
  106. Polygon(E_1, H_1, G_1) → poly4, g_4, e_4, h_4
  107. Line(C_1, D_1) → l_3
  108. Line(G_1, H_1) → m_3
  109. Angle(l_3, m_3) → θ
  110. Segment(D_1, H_1) → n_3
  111. Line(D_1, H_1) → p_3
  112. Circle(D_1, (u * a_3)) → q_3
  113. Intersect(q_3, p_3) → I_1, J_1
  114. Vector(D_1, J_1) → w_5
  115. Translate(B_1, w_5) → K_1
  116. Translate(D_1, w_5) → L_1
  117. Translate(C_1, w_5) → O_1
  118. Polygon(K_1, L_1, O_1) → poly3', o_1, k_4, l_4
  119. Segment(J_1, O_1) → r_3
  120. Segment(O_1, K_1) → s_3
  121. Segment(K_1, J_1) → t_3
  122. Vector(D_1, H_1) → u_1
  123. Translate(B_1, u_1) → Q_1
  124. Translate(D_1, u_1) → R_1
  125. Translate(C_1, u_1) → S_1
  126. Polygon(Q_1, R_1, S_1) → poly3'_1, s_4, q_4, r_4
  127. Rotate(Q_1, ((u - 1) * θ), H_1) → T_1
  128. Rotate(R_1, ((u - 1) * θ), H_1) → U_1
  129. Rotate(S_1, ((u - 1) * θ), H_1) → V_1
  130. Polygon(T_1, U_1, V_1) → poly5, v_5, t_4, u_2
  131. Segment(H_1, T_1) → f_4
  132. Segment(T_1, V_1) → i_4
  133. Segment(V_1, H_1) → j_4
  134. Angle(K_1, J_1, O_1) → η
  135. Angle(T_1, H_1, V_1) → ι
  136. Angle(E_1, H_1, G_1) → κ
  137. Polygon(E, U, T) → poly6, t_5, e_5, u_3
  138. Semicircle(W_1, Z_1) → c_4
  139. Point(c_4) → C_2
  140. Polygon(W_1, C_2, Z_1) → poly7, z_1, w_1, c_5
  141. Point(Circle[D_2, c_5]) → E_2
  142. Segment(D_2, E_2) → m_4
  143. Semicircle(D_2, E_2) → d_4
  144. Circle(D_2, z_1) → p_4
  145. Intersect(d_4, p_4, 1) → F_2
  146. Polygon(D_2, F_2, E_2) → poly8, e_6, d_5, f_5
  147. Segment(W_1, Z_1) → n_4
  148. Segment(Z_1, E_2) → b_4
  149. Circle(Z_1, (v * b_4)) → g_5
  150. Vector(Z_1, E_2) → w_2
  151. Translate(W_1, w_2) → J_2
  152. Translate(Z_1, w_2) → K_2
  153. Vector(Z_1, E_2) → v_3
  154. Translate(W_1, v_3) → L_2
  155. Translate(Z_1, v_3) → O_2
  156. Segment(L_2, O_2) → h_5
  157. Angle(L_2, E_2, D_2) → λ
  158. Rotate(L_2, ((v - 1) * λ), E_2) → R_2
  159. Rotate(O_2, ((v - 1) * λ), E_2) → S_2
  160. Segment(R_2, S_2) → i_5
  161. Point(poly7) → U_2
  162. CircleSector(W_1, U_2, C_2) → k_5
  163. Segment(W_1, U_2) → j_5
  164. Circle(D_2, j_5) → p_5
  165. Intersect(m_4, f_5) → V_2
  166. Intersect(m_4, f_5) → W_2
  167. Intersect(m_4, f_5) → Z_2
  168. Intersect(p_5, f_5) → A_3, J_3
  169. CircleSector(D_2, A_3, F_2) → q_5
  170. Segment(E_2, D_2) → m_5
  171. Vector(W_1, D_2) → u_4
  172. Translate(k_5, u_4) → c_6
  173. Rotate(c_6, ((v - 3) * λ), D_2) → d_6
  174. Angle(W_1, C_2, Z_1) → μ
  175. Angle(D_2, F_2, E_2) → ν
  176. Segment(W_1, D_2) → n_5
  177. Line(W_1, D_2) → s_5
  178. Point(Segment[W_1, D_2], v - 2) → A_4
  179. Vector(W_1, A_4) → w_3
  180. Translate(k_5, w_3) → r_5
  181. Midpoint(i_5) → B_3
  182. Midpoint(m_4) → D_3
  183. Point(r_5) → E_3
  184. Point(d_6) → F_3
  185. Angle(J_1, O_1, K_1) → ξ
  186. Angle(H_1, G_1, E_1) → ο
  187. Angle(H_1, V_1, T_1) → ρ
  188. Point(Segment[Z_1, E_2], v) → A_5
  189. Vector(Z_1, A_5) → v_2
  190. Translate(W_1, v_2) → G_2
  191. Translate(Z_1, v_2) → H_2
  192. Segment(G_2, H_2) → a_4
  193. Midpoint(a_4) → I_2
  194. Line(D_2, c_5) → a_5
  195. Circle(D_2, E_2) → f_6
  196. Intersect(f_6, a_5, 1) → T_2
  197. Ray(D_2, F_2) → b_5
  198. Point(b_5) → C_3
  199. Point(Segment[D_2, C_3], v - 4) → A_6
  200. Segment(D_2, A_6) → g_6
  201. Vector(W_1, A_4) → v_4
  202. Translate(k_5, v_4) → h_6
  203. Point(Segment[S_2, F_2], v - 5) → A_7
  204. Segment(S_2, A_7) → i_6
  205. Point(q_5) → G_3
  206. Angle(D_2, F_2, E_2) → σ
  207. Point(Segment[D_2, F_2], v - 6) → A_8
  208. Segment(D_2, A_8) → j_6
  209. Polygon(W_1, C_2, Z_1) → poly9, z_2, w_4, c_7
  210. Polygon(D_2, F_2, E_2) → poly10, e_7, d_7, f_7
  211. Rotate(Z_1, λ, W_1) → I_3
  212. Circle(W_1, I_3) → k_6
  213. Rotate(D_2, (-(((v - 8) * λ))), W_1) → K_3
  214. Rotate(F_2, (-(((v - 8) * λ))), W_1) → L_3
  215. Rotate(E_2, (-(((v - 8) * λ))), W_1) → M_3
  216. Polygon(K_3, L_3, M_3) → poly10', m_6, k_7, l_6
  217. Segment(S_2, D_2) → n_6
  218. Segment(W_1, M_3) → p_6
  219. Segment(L_3, M_3) → q_6
  220. Segment(L_3, W_1) → r_6
  221. Angle(W_1, L_3, M_3) → τ
  222. Angle(M_3, W_1, L_3) → υ
  223. Segment(S_3, O_3) → b_6
  224. Circle(Q_3, s_7) → t_6
  225. Point(t_6) → T_3
  226. Semicircle(Q_3, T_3) → g_7
  227. Circle(Q_3, r_7) → h_7
  228. Intersect(g_7, h_7, 1) → V_3
  229. Polygon(Q_3, V_3, T_3) → poly12, t_7, q_7, v_1
  230. Point(Segment[S_3, V_3], vo) → B_4
  231. Vector(S_3, B_4) → u_5
  232. Translate(O_3, u_5) → W_3
  233. Translate(S_3, u_5) → Z_3
  234. Segment(W_3, Z_3) → a_6
  235. Line(S_3, t_7) → i_7
  236. Intersect(i_7, s_7) → C_4
  237. Angle(O_3, S_3, C_4) → ϕ
  238. Rotate(W_3, ((vo - 1) * ϕ), V_3) → D_4
  239. Rotate(Z_3, ((vo - 1) * ϕ), V_3) → E_4
  240. Segment(D_4, E_4) → j_7
  241. Ray(V_3, T_3) → l_7
  242. Point(l_7) → F_4
  243. Point(Segment[V_3, F_4], vo - 2) → B_5
  244. Segment(V_3, B_5) → m_7
  245. Angle(Q_3, V_3, T_3) → χ
  246. Point(Segment[O_3, Q_3], vo - 3) → B_6
  247. Vector(O_3, B_6) → u_6
  248. Translate(R_3, u_6) → G_4
  249. Translate(O_3, u_6) → H_4
  250. Segment(G_4, H_4) → n_7
  251. Line(Q_3, s_7) → p_7
  252. Intersect(t_6, p_7, 1) → I_4
  253. Circle(T_3, I_4) → c_8
  254. Intersect(t_6, c_8, 2) → J_4
  255. CircleArc(Q_3, I_4, J_4) → d_8
  256. Point(d_8, vo - 4) → B_7
  257. Segment(Q_3, B_7) → a_7
  258. CircleArc(Q_3, I_4, B_7) → e_8
  259. Point(Segment[Q_3, T_3], vo - 5) → B_8
  260. Segment(Q_3, B_8) → b_7
  261. Point(Segment[V_3, T_3], vo - 6) → B_9
  262. Segment(V_3, B_9) → f_8
  263. Point(Segment[V_3, S_3], vot) → C_5
  264. Vector(V_3, C_5) → v_6
  265. Translate(Q_3, v_6) → K_4
  266. Translate(V_3, v_6) → L_4
  267. Translate(T_3, v_6) → M_4
  268. Polygon(K_4, L_4, M_4) → poly12', m_8, k_8, l_8
  269. Rotate(K_4, (-(((vot - 1) * ϕ))), C_5) → O_4
  270. Rotate(L_4, (-(((vot - 1) * ϕ))), C_5) → Q_4
  271. Rotate(M_4, (-(((vot - 1) * ϕ))), C_5) → R_4
  272. Polygon(O_4, Q_4, R_4) → poly12'', r_8, o_4, q_8
  273. Segment(C_5, O_4) → g_8
  274. Angle(O_3, S_3, R_3) → ψ
  275. Angle(K_4, C_5, M_4) → ω
  276. Angle(O_4, C_5, R_4) → α_1
  277. Midpoint(a_6) → S_4
  278. Midpoint(t_7) → T_4
  279. Midpoint(H_4, G_4) → U_4
  280. Segment(Q_3, T_3) → h_8
  281. Midpoint(Q_3, T_3) → V_4
  282. Line(W_4, Z_4) → i_8
  283. Line(D_5, i_8) → j_8
  284. Segment(Z_4, D_5) → n_8
  285. Line(W_4, n_8) → p_8
  286. Intersect(j_8, p_8) → E_5
  287. Polygon(W_4, Z_4, D_5, E_5) → poly13, w_6, z_4, d_9, e_9
  288. Line(E_5, Z_4) → s_8
  289. Segment(W_4, D_5) → t_8
  290. OrthogonalLine(W_4, s_8) → a_8
  291. OrthogonalLine(D_5, s_8) → b_8
  292. Intersect(s_8, a_8) → F_5
  293. Intersect(s_8, b_8) → G_5
  294. Segment(E_5, Z_4) → f_9
  295. Intersect(t_8, f_9) → H_5
  296. Polygon(W_4, F_5, H_5) → poly14, h_9, w_7, f_{10}
  297. Polygon(D_5, H_5, G_5) → poly15, g_9, d_{10}, h_{10}
  298. Rotate(W_4, (-((ap12 * π))), H_5) → I_5
  299. Rotate(F_5, (-((ap12 * π))), H_5) → J_5
  300. Rotate(H_5, (-((ap12 * π))), H_5) → K_5
  301. Polygon(I_5, J_5, K_5) → poly14', k_9, i_9, j_9
  302. Angle(H_5, F_5, W_4) → β_1
  303. Angle(H_5, G_5, D_5) → γ_1
  304. Circle(M_5, O_5) → c_9
  305. Point(c_9) → P_5
  306. Polygon(M_5, O_5, P_5) → poly16, p_9, m_9, o_5
  307. OrthogonalLine(O_5, p_9) → l_9
  308. OrthogonalLine(P_5, o_5) → n_9
  309. Intersect(l_9, n_9) → Q_5
  310. Polygon(M_5, O_5, Q_5) → poly17, q_9, m_{10}, o_6
  311. Polygon(Q_5, P_5, M_5) → poly18, m_{11}, q_{10}, p_{10}
  312. Segment(O_5, P_5) → r_9
  313. Ray(P_5, Q_5) → s_9
  314. Point(s_9) → R_5
  315. Segment(P_5, R_5) → t_9
  316. Ray(O_5, Q_5) → a_9
  317. Point(a_9) → S_5
  318. Segment(O_5, S_5) → b_9
  319. Angle(Q_5, P_5, M_5) → δ_1
  320. Angle(M_5, O_5, Q_5) → ε_1
  321. Rotate(O_5, (ap22 * π), O_5) → T_5
  322. Rotate(Q_5, (ap22 * π), O_5) → U_5
  323. Segment(O_5, U_5) → i_{10}
  324. Mirror(Q_5, O_5) → V_5
  325. Segment(O_5, V_5) → g_{10}
  326. Polygon(M_5, O_5, V_5) → poly19, v_7, m_{12}, o_7
  327. Point(Segment[M_5, V_5], ap22 - 1) → C_6
  328. Segment(M_5, C_6) → j_{10}
  329. Angle(P_5, M_5, Q_5) → ζ_1
  330. Rotate(M_5, (((ap22 - 1) * 2) * ζ_1), M_5) → W_5
  331. Rotate(M_5, (-((((ap22 - 1) * 2) * ζ_1))), M_5) → Z_5
  332. Rotate(M_5, (-((((ap22 - 2) * 2) * ζ_1))), M_5) → D_6
  333. Rotate(O_5, (-((((ap22 - 3) * 2) * ζ_1))), M_5) → E_6
  334. Rotate(V_5, (-((((ap22 - 3) * 2) * ζ_1))), M_5) → F_6
  335. Polygon(D_6, E_6, F_6) → poly19', f_{11}, d_{11}, e_{10}
  336. Segment(M_5, E_6) → k_{10}
  337. Angle(K_5, J_5, I_5) → η_1

AI Drawing Prompt Reference

Draw a composite figure of multiple right triangles that can be superimposed through transformations. Start with a semicircle, pick a point on it, and form a right triangle with the semicircle's endpoints. Then create another semicircle and a circle, intersect them to get a point, and form a second right triangle. Next, draw a segment, use it as a radius for a circle, intersect with a line to get a point, and form a third right triangle. Then draw two intersecting segments, use one as a radius for a circle, intersect with the other segment, and form a fourth right triangle. Finally, draw two concentric circles, intersect them with segments to form a fifth right triangle. All triangles are congruent and can be superimposed via translations, rotations, or reflections.
The prompt above was compiled from the original drawing commands and is for reference only.