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Proof Without Words by Steve Phelps (4)

Proof Without Words by Steve Phelps (4)
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2015-02-18 07:57
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Drag points and sliders to observe changes and explore geometric proofs. This example demonstrates proving geometric properties through geometric transformations. It uses commands like constructing polygons, circles, vectors, and transformations in GeoGebra. Suitable for middle school and above students learning geometric proofs.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  4. Point(xAxis) → E
  5. Polygon(E, F, 4) → poly2, e, f, g, h, G, H
  6. Point(d) → I
  7. OrthogonalLine(I, d) → i
  8. Circle(C, Segment[A, I]) → k
  9. Intersect(k, c, 1) → J
  10. OrthogonalLine(J, c) → j
  11. Intersect(i, j) → K
  12. Segment(A, K) → l
  13. Segment(K, C) → m
  14. Intersect(j, a) → L
  15. Intersect(i, b) → M
  16. Incircle(J, C, K) → p
  17. Incircle(C, M, K) → q
  18. Incircle(K, L, A) → r
  19. Incircle(I, K, A) → s
  20. Center(s) → N
  21. Center(p) → O
  22. Center(q) → P
  23. Center(r) → Q
  24. Intersect(s, i, 1) → R
  25. Intersect(s, d, 1) → S
  26. Intersect(s, l, 1) → T
  27. Intersect(r, l, 1) → U
  28. Intersect(r, j, 1) → V
  29. Intersect(r, a, 1) → W
  30. Intersect(q, m, 1) → Z
  31. Intersect(q, i, 1) → A_1
  32. Intersect(q, b, 1) → B_1
  33. Intersect(p, m, 1) → C_1
  34. Intersect(p, j, 1) → D_1
  35. Segment(D_1, O) → t
  36. Segment(O, C_1) → a_1
  37. Segment(Z, P) → b_1
  38. Segment(P, B_1) → c_1
  39. Segment(P, A_1) → d_1
  40. Segment(U, Q) → e_1
  41. Segment(Q, V) → f_1
  42. Segment(Q, W) → g_1
  43. Segment(T, N) → h_1
  44. Segment(N, S) → i_1
  45. Segment(N, R) → j_1
  46. Intersect(p, c) → E_1, F_1
  47. Segment(O, E_1) → n
  48. Segment(J, K) → k_1
  49. Segment(K, L) → l_1
  50. Segment(K, M) → m_1
  51. Segment(K, I) → n_1
  52. Polygon(I, S, N, R) → poly3, i_2, s_1, n_2, r_1
  53. Polygon(Q, W, L, V) → poly4, q_1, w, l_2, v
  54. Polygon(D_1, J, E_1, O) → poly5, d_2, j_2, e_2, o
  55. Polygon(P, A_1, M, B_1) → poly6, p_1, a_2, m_2, b_2
  56. If(0 ≤ zz ≤ 1, 1 - zz, If[zz > 1, 0, 1]) → d1
  57. Dilate(M, d1, F) → M'
  58. Vector(M, M') → u
  59. Translate(P, u) → P'
  60. Translate(A_1, u) → G_1
  61. Translate(M, u) → M'_1
  62. Translate(B_1, u) → H_1
  63. Polygon(P', G_1, M'_1, H_1) → poly6', p', g_2, m'_1, h_2
  64. Dilate(O, d1, H_1) → O'
  65. Dilate(L, d1, G_1) → L'
  66. Dilate(N, d1, P') → N'
  67. Vector(O, O') → u_1
  68. Vector(L, L') → v_1
  69. Vector(N, N') → w_1
  70. Translate(D_1, u_1) → I_1
  71. Translate(J, u_1) → J'
  72. Translate(E_1, u_1) → J_1
  73. Translate(O, u_1) → O'_1
  74. Polygon(I_1, J', J_1, O'_1) → poly5', i_3, j', j_3, o'_1
  75. Translate(Q, v_1) → Q'
  76. Translate(W, v_1) → W'
  77. Translate(L, v_1) → L'_1
  78. Translate(V, v_1) → V'
  79. Polygon(Q', W', L'_1, V') → poly4', q', w', l'_1, v'
  80. Translate(I, w_1) → I'
  81. Translate(S, w_1) → S'
  82. Translate(N, w_1) → N'_1
  83. Translate(R, w_1) → R'
  84. Polygon(I', S', N'_1, R') → poly3', i', s', n'_1, r'
  85. Segment(P, C) → t_1
  86. Segment(O, K) → c_2
  87. Segment(K, P) → f_2
  88. Segment(Q, K) → k_2
  89. Segment(Q, A) → p_2
  90. Segment(N, A) → q_2
  91. Segment(N, K) → r_2
  92. Polygon(B_1, C, P) → poly7, p_3, b_3, c_3
  93. If(1 ≤ zz ≤ 2, 2 - zz, If[zz > 2, 0, 1]) → d2
  94. Dilate(B_1, d2, J_1) → K_1
  95. Vector(B_1, K_1) → u_2
  96. Translate(B_1, u_2) → L_1
  97. Translate(C, u_2) → C'
  98. Translate(P, u_2) → P'_1
  99. Polygon(L_1, C', P'_1) → poly7', p'_1, l_3, c'
  100. Polygon(K, D_1, O) → poly8, o_1, k_3, d_3
  101. Dilate(O, d2, C') → O'_2
  102. Vector(O, O'_2) → v_2
  103. Translate(K, v_2) → K'
  104. Translate(D_1, v_2) → M_1
  105. Translate(O, v_2) → O'_3
  106. Polygon(K', M_1, O'_3) → poly8', o'_3, k', m_3
  107. Dilate(W, d2, W') → W'_1
  108. Vector(W, W'_1) → w_2
  109. Polygon(A, Q, W) → poly9, w_3, a_3, q_3
  110. Translate(A, w_2) → A'
  111. Translate(Q, w_2) → Q'_1
  112. Translate(W, w_2) → W'_2
  113. Polygon(A', Q'_1, W'_2) → poly9', w'_2, a', q'_1
  114. Polygon(N, R, K) → poly10, k_4, n_3, r_3
  115. Dilate(K, d2, Q'_1) → K'_1
  116. Vector(K, K'_1) → s_2
  117. Translate(N, s_2) → N'_2
  118. Translate(R, s_2) → R'_1
  119. Translate(K, s_2) → K'_2
  120. Polygon(N'_2, R'_1, K'_2) → poly10', k'_2, n'_2, r'_1
  121. Angle(W, A, T) → α
  122. Polygon(T, N, K) → poly11, k_5, t_2, n_4
  123. Translate(T, s_2) → T'
  124. Translate(N, s_2) → N'_3
  125. Translate(K, s_2) → K'_3
  126. Polygon(T', N'_3, K'_3) → poly11', k'_3, t', n'_3
  127. If(1 ≤ zz ≤ 2, zz - 1, If[zz > 2, 1, 0]) → r1
  128. Rotate(T', (-((α * r1))), K'_1) → T''
  129. Rotate(N'_3, (-((α * r1))), K'_1) → N_1
  130. Rotate(K'_3, (-((α * r1))), K'_1) → O_1
  131. Polygon(T'', N_1, O_1) → poly11'', o_2, t'', n_5
  132. Polygon(A, U, Q) → poly12, q_4, a_4, u_3
  133. Dilate(Q, d2, Q') → Q'_2
  134. Vector(Q, K'_1) → v_3
  135. Translate(A, v_3) → A'_1
  136. Translate(U, v_3) → U'
  137. Translate(Q, v_3) → Q'_3
  138. Rotate(A'_1, (-((α * r1))), K'_1) → P_1
  139. Rotate(U', (-((α * r1))), K'_1) → U''
  140. Rotate(Q'_3, (-((α * r1))), K'_1) → Q_1
  141. Dilate(Q, d2, Q') → Q'_4
  142. Vector(Q, Q'_1) → e_3
  143. Translate(A, e_3) → A'_2
  144. Translate(U, e_3) → U'_1
  145. Translate(Q, e_3) → Q'_5
  146. Polygon(A'_2, U'_1, Q'_5) → poly12', q'_5, a'_2, u'_1
  147. Rotate(A'_2, (-((α * r1))), Q'_1) → R_1
  148. Rotate(U'_1, (-((α * r1))), Q'_1) → S_1
  149. Rotate(Q'_5, (-((α * r1))), Q'_1) → T_1
  150. Polygon(R_1, S_1, T_1) → poly12'', t_3, r_4, s_3
  151. Polygon(K, O, C_1) → poly13, c_4, k_6, o_3
  152. Dilate(K, d2, K') → K'_4
  153. Vector(K, K'_4) → f_3
  154. Translate(K, f_3) → K'_5
  155. Translate(O, f_3) → O'_4
  156. Translate(C_1, f_3) → U_1
  157. Polygon(K'_5, O'_4, U_1) → poly13', u_4, k'_5, o'_4
  158. Rotate(K'_5, (α * r1), K'_4) → V_1
  159. Rotate(O'_4, (α * r1), K'_4) → W_1
  160. Rotate(U_1, (α * r1), K'_4) → Z_1
  161. Polygon(V_1, W_1, Z_1) → poly13'', z_1, v_4, w_4
  162. Polygon(C, Z, P) → poly14, p_4, c_5, z_2
  163. Dilate(P, d2, K'_4) → P'_2
  164. Vector(P, P'_2) → g_3
  165. Translate(C, g_3) → C'_1
  166. Translate(Z, g_3) → Z'
  167. Translate(P, g_3) → P'_3
  168. Polygon(C'_1, Z', P'_3) → poly14', p'_3, c'_1, z'
  169. Rotate(C'_1, (α * r1), P'_2) → A_2
  170. Rotate(Z', (α * r1), P'_2) → Z''
  171. Rotate(P'_3, (α * r1), P'_2) → B_2
  172. Polygon(A_2, Z'', B_2) → poly14'', b_4, a_5, z''
  173. Segment(O, C) → h_3
  174. Polygon(A, S, N) → poly15, n_6, a_6, s_4
  175. Polygon(A, T, N) → poly16, n_7, a_7, t_4
  176. Polygon(Q, U, K) → poly17, k_7, q_5, u_5
  177. Polygon(Q, V, K) → poly18, k_8, q_6, v_5
  178. If(2 ≤ zz ≤ 3, 3 - zz, If[zz > 2, 0, 1]) → d3
  179. If(2 ≤ zz ≤ 3, zz - 2, If[zz > 3, 1, 0]) → r2
  180. Dilate(V, d3, E) → V'_1
  181. Vector(V, V'_1) → d_4
  182. Translate(Q, d_4) → Q'_6
  183. Translate(V, d_4) → V'_2
  184. Translate(K, d_4) → K'_6
  185. Polygon(Q'_6, V'_2, K'_6) → poly18', k'_6, q'_6, v'_2
  186. Rotate(Q'_6, (-((90° * r2))), V'_1) → C_2
  187. Rotate(V'_2, (-((90° * r2))), V'_1) → D_2
  188. Rotate(K'_6, (-((90° * r2))), V'_1) → E_2
  189. Polygon(C_2, D_2, E_2) → poly18'', e_4, c_6, d_5
  190. Dilate(N, d3, A') → N'_4
  191. Vector(N, N'_4) → f_4
  192. Translate(A, f_4) → A'_3
  193. Translate(S, f_4) → S'_1
  194. Translate(N, f_4) → N'_5
  195. Polygon(A'_3, S'_1, N'_5) → poly15', n'_5, a'_3, s'_1
  196. Rotate(A'_3, (-(((r2 * 90))°)), N'_4) → F_2
  197. Rotate(S'_1, (-(((r2 * 90))°)), N'_4) → G_2
  198. Rotate(N'_5, (-(((r2 * 90))°)), N'_4) → H_2
  199. Polygon(F_2, G_2, H_2) → poly15'', h_4, f_5, g_4
  200. Dilate(T, d3, R'_1) → T'_1
  201. Vector(T, T'_1) → i_4
  202. Translate(A, i_4) → A'_4
  203. Translate(T, i_4) → T'_2
  204. Translate(N, i_4) → N'_6
  205. Polygon(A'_4, T'_2, N'_6) → poly16', n'_6, a'_4, t'_2
  206. Rotate(A'_4, (-((r2 * α))), T'_1) → I_2
  207. Rotate(T'_2, (-((r2 * α))), T'_1) → J_2
  208. Rotate(N'_6, (-((r2 * α))), T'_1) → K_2
  209. Polygon(I_2, J_2, K_2) → poly16'', k_9, i_5, j_4
  210. Dilate(K, d3, N_1) → K'_7
  211. Vector(K, K'_7) → l_4
  212. Translate(Q, l_4) → Q'_7
  213. Translate(U, l_4) → U'_2
  214. Translate(K, l_4) → K'_8
  215. Polygon(Q'_7, U'_2, K'_8) → poly17', k'_8, q'_7, u'_2
  216. Rotate(Q'_7, (-((r2 * α))), K'_7) → L_2
  217. Rotate(U'_2, (-((r2 * α))), K'_7) → M_2
  218. Rotate(K'_8, (-((r2 * α))), K'_7) → N_2
  219. Polygon(L_2, M_2, N_2) → poly17'', n_8, l_5, m_4
  220. Polygon(O, C, C_1) → poly19, c_7, o_4, c_8
  221. Polygon(O, E_1, C) → poly20, c_9, o_5, e_5
  222. Polygon(K, Z, P) → poly21, p_5, k_{10}, z_3
  223. Polygon(K, A_1, P) → poly22, p_6, k_{11}, a_8
  224. Dilate(C, d3, G) → C'_2
  225. Vector(C, C'_2) → w_5
  226. Translate(O, w_5) → O'_5
  227. Translate(C, w_5) → C'_3
  228. Translate(C_1, w_5) → O_2
  229. Polygon(O'_5, C'_3, O_2) → poly19', o_6, o'_5, c'_3
  230. Rotate(O'_5, (r2 * α), C'_2) → P_2
  231. Rotate(C'_3, (r2 * α), C'_2) → Q_2
  232. Rotate(O_2, (r2 * α), C'_2) → R_2
  233. Polygon(P_2, Q_2, R_2) → poly19'', r_5, p_7, q_7
  234. Dilate(P, d3, G) → P'_4
  235. Vector(P, P'_4) → b_5
  236. Translate(K, b_5) → K'_9
  237. Translate(Z, b_5) → Z'_1
  238. Translate(P, b_5) → P'_5
  239. Polygon(K'_9, Z'_1, P'_5) → poly21', p'_5, k'_9, z'_1
  240. Rotate(K'_9, (r2 * α), P'_4) → S_2
  241. Rotate(Z'_1, (r2 * α), P'_4) → T_2
  242. Rotate(P'_5, (r2 * α), P'_4) → U_2
  243. Polygon(S_2, T_2, U_2) → poly21'', u_6, s_5, t_5
  244. Dilate(P, d3, M_1) → P'_6
  245. Vector(P, P'_6) → g_5
  246. Translate(K, g_5) → K'_{10}
  247. Translate(A_1, g_5) → V_2
  248. Translate(P, g_5) → P'_7
  249. Polygon(K'_{10}, V_2, P'_7) → poly22', p'_7, k'_{10}, v_6
  250. Rotate(K'_{10}, (-(((r2 * 90))°)), P'_6) → W_2
  251. Rotate(V_2, (-(((r2 * 90))°)), P'_6) → Z_2
  252. Rotate(P'_7, (-(((r2 * 90))°)), P'_6) → A_3
  253. Polygon(W_2, Z_2, A_3) → poly22'', a_9, w_6, z_4
  254. Dilate(C, d3, M_1) → C'_4
  255. Vector(C, C'_4) → h_5
  256. Translate(O, h_5) → O'_6
  257. Translate(E_1, h_5) → B_3
  258. Translate(C, h_5) → C'_5
  259. Polygon(O'_6, B_3, C'_5) → poly20', c'_5, o'_6, b_6
  260. Rotate(O'_6, (-(((r2 * 90))°)), C'_4) → C_3
  261. Rotate(B_3, (-(((r2 * 90))°)), C'_4) → D_3
  262. Rotate(C'_5, (-(((r2 * 90))°)), C'_4) → E_3
  263. Polygon(C_3, D_3, E_3) → poly20'', e_6, c_{10}, d_6

AI Drawing Prompt Reference

Draw a quadrilateral and two squares, observing geometric transformations by dragging points and sliders. Start by constructing a quadrilateral and two squares, then connect the diagonals and draw perpendicular lines. Next, draw circles centered at one vertex and the diagonals, intersecting to form incircles. Finally, connect key intersection points to create multiple small triangles and quadrilaterals, demonstrating geometric properties.
The prompt above was compiled from the original drawing commands and is for reference only.