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Dynamic Phenomena with Squares on Triangle Sides

Dynamic Phenomena with Squares on Triangle Sides
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2017-04-30 08:48
TrianglePolygonVectorGeometric Construction
This example dynamically illustrates various geometric phenomena when three squares are constructed on the sides of any triangle. Key math concepts include geometric transformations, vectors, angles, and polygon construction. The construction utilizes commands like Circle, Point, Polygon, Midpoint, Segment, Dilate, Rotate, Min, Intersect, CircleSector, Vector, Translate, Line, Angle, Distance, and OrthogonalLine, as well as objects like c, b, A, d, B, etc. It is suitable for high school and above students to explore patterns in dynamic geometry and transformations.

GeoGebra Commands & Parameters

  1. Circle(A, b) → d
  2. Point(d) → B
  3. Polygon(A, B, 4) → poly1, f, g, h, i, D, E
  4. Circle(B, c) → e
  5. Point(e) → C
  6. Polygon(C, B, 4) → poly2, j, k, l, m, F, G
  7. Midpoint(E, G) → A_2
  8. Segment(A, A_2) → b_4
  9. Midpoint(D, A) → H
  10. Midpoint(F, C) → I
  11. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], A) → B'
  12. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], F) → B'_1
  13. Segment(B, B') → n
  14. Segment(B, B'_1) → p
  15. Rotate(B, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → B'_2
  16. Rotate(B'_1, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → L
  17. Segment(B'_2, L) → p'
  18. Rotate(B, If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B'_3
  19. Rotate(B', If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B''
  20. Segment(B'_3, B'') → n'
  21. Rotate(B'_2, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → M
  22. Rotate(L, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → L'
  23. Segment(M, L') → p''
  24. Rotate(B'_3, If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → N
  25. Rotate(B'', If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → B'''
  26. Segment(N, B''') → n''
  27. Rotate(N, If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → N'
  28. Rotate(B''', If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → O
  29. Segment(N', O) → n'''
  30. Rotate(M, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → M'
  31. Rotate(L', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → L''
  32. Segment(M', L'') → p'''
  33. Segment(F, G) → q
  34. Segment(F, B) → r
  35. Segment(A, B) → s
  36. Min(r, s) → o
  37. Intersect(l, q) → P
  38. Circle(F, (0.3 * b_4)) → t
  39. Intersect(t, q) → Q,
  40. Intersect(t, r) → R,
  41. CircleSector(F, Q, R) → c_1
  42. Dilate(F, If[4 < a ≤ 5, 5 - a, 0], C) → F'
  43. Vector(F, F') → u
  44. Translate(c_1, u) → d_1
  45. Rotate(d_1, If[5 < a ≤ 6, ((a - 5) * π), π], C) → e_1
  46. Dilate(F, If[6 < a ≤ 7, 7 - a, 0], A) → F'_1
  47. Dilate(C, If[6 < a ≤ 7, 7 - a, 0], E) → C'
  48. Vector(F, F'_1) → v
  49. Vector(C, C') → w
  50. Translate(e_1, w) → f_1
  51. Translate(c_1, v) → g_1
  52. Line(A, p) → h_1
  53. Line(E, p''') → i_1
  54. Point(h_1) → S
  55. Angle(S, A, E) → α
  56. Rotate(g_1, If[7 < a ≤ 8, ((-(a - 7)) * ((2 * π) - α)), (-((2 * π) - α))], A) → k_1
  57. Line(E, h_1) → j_1
  58. Point(j_1) → T
  59. Angle(D, E, T) → β
  60. Rotate(f_1, If[7 < a ≤ 8, ((-(a - 7)) * β), (-β)], E) → p_1
  61. Dilate(E, If[8 < a ≤ 9, 9 - a, 0], G) → E'
  62. Dilate(F, If[8 < a ≤ 9, 9 - a, 0], D) → F'_2
  63. Vector(E, E') → u_1
  64. Vector(F, F'_2) → v_1
  65. Translate(p_1, u_1) → q_1
  66. Translate(c_1, v_1) → r_1
  67. Line(G, n'') → l_1
  68. Point(l_1) → U
  69. Line(G, n') → m_1
  70. Point(m_1) → V
  71. Angle(C, G, V) → γ
  72. Rotate(q_1, If[8 < a ≤ 9, ((-(a - 8)) * γ), (-γ)], E') → s_1
  73. Line(D, p) → n_1
  74. Point(n_1) → W
  75. Angle(W, D, B) → δ
  76. Rotate(r_1, If[8 < a ≤ 9, ((a - 8) * δ), δ], F'_2) → t_1
  77. Midpoint(A, B) → Z
  78. Midpoint(B, C) → A_1
  79. Midpoint(Z, B) → B_1
  80. Midpoint(A_1, B) → C_1
  81. Dilate(C, If[9 < a ≤ 10, 10 - a, 0], C_1) → C'_1
  82. Dilate(A, If[9 < a ≤ 10, 10 - a, 0], B_1) → A'
  83. Vector(A, A') → w_1
  84. Vector(C, C'_1) → a_1
  85. Translate(e_1, a_1) → c_2
  86. Translate(k_1, w_1) → d_2
  87. Rotate(d_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], B_1) → e_2
  88. Rotate(c_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], C_1) → f_2
  89. Dilate(C_1, If[11 < a ≤ 12, 12 - a, 0], B) → D_1
  90. Dilate(B_1, If[11 < a ≤ 12, 12 - a, 0], B) → E_1
  91. Vector(B_1, E_1) → b_1
  92. Vector(C_1, D_1) → u_2
  93. Translate(e_2, b_1) → g_2
  94. Translate(f_2, u_2) → h_2
  95. Dilate(F, If[12 < a ≤ 13, 13 - a, 0], I) → F'_3
  96. Segment(F, F'_3) → i_2
  97. Dilate(D, If[12 < a ≤ 13, 13 - a, 0], H) → D'
  98. Segment(D, D') → j_2
  99. Rotate(F, If[13 < a ≤ 14, ((a - 13) * π), π], I) → F'_4
  100. Rotate(F'_3, If[13 < a ≤ 14, ((a - 13) * π), π], I) → F_1
  101. Segment(F'_4, F_1) → k_2
  102. Rotate(D, If[13 < a ≤ 14, ((a - 13) * π), π], H) → D'_1
  103. Rotate(D', If[13 < a ≤ 14, ((a - 13) * π), π], H) → D''
  104. Segment(D'_1, D'') → l_2
  105. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], I) → B'_4
  106. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], H) → B'_5
  107. Vector(B, B'_4) → v_2
  108. Vector(B, B'_5) → w_2
  109. Line(I, p) → g_3
  110. Point(g_3) → N_1
  111. Line(H, n''') → l_3
  112. Point(l_3) → O_1
  113. Polygon(A, F, B) → poly3, b_2, a_2, f_3
  114. Circle(A, 2) → c_4
  115. Circle(F, 2) → d_4
  116. Polygon(F, A, 4) → poly4, i_4, j_4, k_4, l_4, C_2, D_2
  117. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], C_2) → A'_1
  118. Segment(A, A'_1) → m_2
  119. Rotate(A, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], C_2) → A'_2
  120. Rotate(A'_1, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], C_2) → J
  121. Segment(A'_2, J) → n_2
  122. Rotate(A'_2, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], D_2) → K
  123. Rotate(J, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], D_2) → J'
  124. Segment(K, J') → p_2
  125. Rotate(K, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], F) → K'
  126. Rotate(J', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], F) → J''
  127. Segment(K', J'') → q_2
  128. Angle(B, F, D_2) → ε
  129. Rotate(c_1, If[12 < a ≤ 13, ((a - 12) * ε), ε], F) → r_2
  130. Midpoint(A, D_2) → G_1
  131. Dilate(A, If[12 < a ≤ 13, 13 - a, 0], G_1) → A'_3
  132. Segment(A, A'_3) → s_2
  133. Rotate(A, If[13 < a ≤ 14, ((a - 13) * π), π], G_1) → A'_4
  134. Rotate(A'_3, If[13 < a ≤ 14, ((a - 13) * π), π], G_1) → H_1
  135. Segment(A'_4, H_1) → t_2
  136. Line(G_1, B) → h_3
  137. Line(A, I) → i_3
  138. Line(F, H) → j_3
  139. Line(H, I) → k_3
  140. Distance(H, I) → distanceHI
  141. Distance(B, G_1) → distanceBG1
  142. Dilate(G_1, If[14 < a ≤ 15, 15 - a, 0], B) → J_1
  143. Dilate(A, If[15 < a ≤ 16, 16 - a, 0], I) → A'_5
  144. Line(H, I) → r_3
  145. Line(B, G_1) → s_3
  146. Intersect(r_3, s_3) → L_1
  147. OrthogonalLine(C, r_3) → t_3
  148. Intersect(t_3, r_3) → M_1
  149. Line(H, I) → a_3
  150. Line(G_1, a_3) → b_3
  151. Rotate(G_1, If[18 < a ≤ 19, ((-(a - 18)) * π / 2), ((-π)) / 2], G_1) → P_1
  152. Dilate(C, If[20 < a ≤ 21, 21 - a, 0], M_1) → C'_2
  153. Vector(C, C'_2) → v_3
  154. Dilate(M_1, If[22 < a ≤ 23, 23 - a, 0], L_1) → V_1
  155. Vector(M_1, V_1) → w_3
  156. Circle(B, 1) → p_4
  157. Circle(G_1, 2) → q_4
  158. Intersect(q_4, s_3, 2) → Z_1
  159. Dilate(E, If[14 < a ≤ 15, 15 - a, 0], A_2) → E'_1
  160. Segment(E, E'_1) → m_3
  161. Dilate(A_2, If[15 < a ≤ 16, 16 - a, 0], G) → B_2
  162. Segment(A_2, B_2) → n_3
  163. Rotate(E, If[16 < a ≤ 17, ((a - 16) * π), π], A_2) → E'_2
  164. Rotate(E'_1, If[16 < a ≤ 17, ((a - 16) * π), π], A_2) → E_2
  165. Segment(E'_2, E_2) → r_4
  166. Dilate(A, If[17 < a ≤ 18, 18 - a, 0], A_2) → A'_6
  167. Segment(A, A'_6) → p_3
  168. Rotate(A, If[18 < a ≤ 19, ((a - 18) * π / 2), π / 2], A_2) → A'_7
  169. Rotate(A'_6, If[18 < a ≤ 19, ((a - 18) * π / 2), π / 2], A_2) → I_1
  170. Segment(A'_7, I_1) → q_3
  171. Rotate(A'_7, If[19 < a ≤ 20, ((a - 19) * π / 2), π / 2], F) → K_1
  172. Rotate(I_1, If[19 < a ≤ 20, ((a - 19) * π / 2), π / 2], F) → Q_1
  173. Segment(K_1, Q_1) → c_3
  174. Rotate(K_1, If[20 < a ≤ 21, ((a - 20) * π / 2), π / 2], G_1) → R_1
  175. Rotate(Q_1, If[20 < a ≤ 21, ((a - 20) * π / 2), π / 2], G_1) → S_1
  176. Segment(R_1, S_1) → d_3
  177. Polygon(A, A_2, F, G_1) → poly5, a_4, a_5, f_4, g_4
  178. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], A_2) → B'_6
  179. Vector(B, B'_6) → u_3
  180. Translate(g_2, u_3) → e_3
  181. Line(B, q_3) → h_4
  182. Point(h_4) → T_1
  183. Angle(A, B, T_1) → ζ
  184. Rotate(e_3, If[21 < a ≤ 22, ((a - 21) * ζ), ζ], B'_6) → e_4
  185. Dilate(D, If[22 < a ≤ 23, 23 - a, 0], A_2) → D'_2
  186. Segment(D, D'_2) → m_4
  187. Rotate(D, If[23 < a ≤ 24, ((-(a - 23)) * π / 2), ((-π)) / 2], A_2) → D'_3
  188. Rotate(D'_2, If[23 < a ≤ 24, ((-(a - 23)) * π / 2), ((-π)) / 2], A_2) → U_1
  189. Segment(D'_3, U_1) → n_4
  190. Dilate(B, If[25 < a ≤ 26, 26 - a, 0], A_2) → B'_7
  191. Vector(B, B'_7) → u_4
  192. Line(B, m_4) → t_4
  193. Point(t_4) → W_1
  194. Angle(C, B, W_1) → κ
  195. Rotate(h_2, If[24 < a ≤ 25, ((a - 24) * κ), κ], B) → s_4
  196. Translate(s_4, u_4) → c_5