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Three Resulting Phenomena from Three Squares Built off the Sides of ANY Triangle

Three Resulting Phenomena from Three Squares Built off the Sides of ANY Triangle
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2017-04-28 05:37
TriangleTransformation
This example illustrates three interesting phenomena when three squares are constructed on the sides of any triangle. Key mathematical concepts include triangles, squares, and geometric transformations. The construction uses GeoGebra commands such as Circle, Polygon, Dilate, etc. Suitable for middle school and above students or teachers.

GeoGebra Commands & Parameters

  1. Circle(A, b) → d
  2. Point(d) → B
  3. Circle(B, c) → e
  4. Point(e) → C
  5. Polygon(A, B, 4) → poly1, f, g, h, i, D, E
  6. Polygon(C, B, 4) → poly2, j, k, l, m, F, G
  7. Midpoint(D, A) → H
  8. Midpoint(F, C) → I
  9. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], A) → B'
  10. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], F) → B'_1
  11. Segment(B, B') → n
  12. Segment(B, B'_1) → p
  13. Rotate(B, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → B'_2
  14. Rotate(B'_1, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → L
  15. Segment(B'_2, L) → p'
  16. Rotate(B, If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B'_3
  17. Rotate(B', If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B''
  18. Segment(B'_3, B'') → n'
  19. Rotate(B'_2, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → M
  20. Rotate(L, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → L'
  21. Segment(M, L') → p''
  22. Rotate(B'_3, If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → N
  23. Rotate(B'', If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → B'''
  24. Segment(N, B''') → n''
  25. Rotate(N, If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → N'
  26. Rotate(B''', If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → O
  27. Segment(N', O) → n'''
  28. Rotate(M, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → M'
  29. Rotate(L', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → L''
  30. Segment(M', L'') → p'''
  31. Segment(F, G) → q
  32. Segment(F, B) → r
  33. Segment(A, B) → s
  34. Min(r, s) → o
  35. Circle(F, (0.15 * o)) → t
  36. Intersect(l, q) → P
  37. Intersect(t, q) → Q,
  38. Intersect(t, r) → R,
  39. CircleSector(F, Q, R) → c_1
  40. Dilate(F, If[4 < a ≤ 5, 5 - a, 0], C) → F'
  41. Vector(F, F') → u
  42. Translate(c_1, u) → d_1
  43. Rotate(d_1, If[5 < a ≤ 6, ((a - 5) * π), π], C) → e_1
  44. Dilate(F, If[6 < a ≤ 7, 7 - a, 0], A) → F'_1
  45. Dilate(C, If[6 < a ≤ 7, 7 - a, 0], E) → C'
  46. Vector(F, F'_1) → v
  47. Vector(C, C') → w
  48. Translate(e_1, w) → f_1
  49. Translate(c_1, v) → g_1
  50. Line(A, p) → h_1
  51. Line(E, p''') → i_1
  52. Point(h_1) → S
  53. Angle(S, A, E) → α
  54. Rotate(g_1, If[7 < a ≤ 8, ((-(a - 7)) * ((2 * π) - α)), (-((2 * π) - α))], A) → k_1
  55. Line(E, h_1) → j_1
  56. Point(j_1) → T
  57. Angle(D, E, T) → β
  58. Rotate(f_1, If[7 < a ≤ 8, ((-(a - 7)) * β), (-β)], E) → p_1
  59. Dilate(E, If[8 < a ≤ 9, 9 - a, 0], G) → E'
  60. Dilate(F, If[8 < a ≤ 9, 9 - a, 0], D) → F'_2
  61. Vector(E, E') → u_1
  62. Vector(F, F'_2) → v_1
  63. Translate(p_1, u_1) → q_1
  64. Translate(c_1, v_1) → r_1
  65. Line(G, n'') → l_1
  66. Point(l_1) → U
  67. Line(G, n') → m_1
  68. Point(m_1) → V
  69. Angle(C, G, V) → γ
  70. Rotate(q_1, If[8 < a ≤ 9, ((-(a - 8)) * γ), (-γ)], E') → s_1
  71. Line(D, p) → n_1
  72. Point(n_1) → W
  73. Angle(W, D, B) → δ
  74. Rotate(r_1, If[8 < a ≤ 9, ((a - 8) * δ), δ], F'_2) → t_1
  75. Midpoint(A, B) → Z
  76. Midpoint(B, C) → A_1
  77. Midpoint(Z, B) → B_1
  78. Midpoint(A_1, B) → C_1
  79. Dilate(C, If[9 < a ≤ 10, 10 - a, 0], C_1) → C'_1
  80. Dilate(A, If[9 < a ≤ 10, 10 - a, 0], B_1) → A'
  81. Vector(A, A') → w_1
  82. Vector(C, C'_1) → a_1
  83. Translate(e_1, a_1) → c_2
  84. Translate(k_1, w_1) → d_2
  85. Rotate(d_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], B_1) → e_2
  86. Rotate(c_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], C_1) → f_2
  87. Dilate(C_1, If[11 < a ≤ 12, 12 - a, 0], B) → D_1
  88. Dilate(B_1, If[11 < a ≤ 12, 12 - a, 0], B) → E_1
  89. Vector(B_1, E_1) → b_1
  90. Vector(C_1, D_1) → u_2
  91. Translate(e_2, b_1) → g_2
  92. Translate(f_2, u_2) → h_2
  93. Dilate(F, If[12 < a ≤ 13, 13 - a, 0], I) → F'_3
  94. Segment(F, F'_3) → i_2
  95. Dilate(D, If[12 < a ≤ 13, 13 - a, 0], H) → D'
  96. Segment(D, D') → j_2
  97. Rotate(F, If[13 < a ≤ 14, ((a - 13) * π), π], I) → F'_4
  98. Rotate(F'_3, If[13 < a ≤ 14, ((a - 13) * π), π], I) → F_1
  99. Segment(F'_4, F_1) → k_2
  100. Rotate(D, If[13 < a ≤ 14, ((a - 13) * π), π], H) → D'_1
  101. Rotate(D', If[13 < a ≤ 14, ((a - 13) * π), π], H) → D''
  102. Segment(D'_1, D'') → l_2
  103. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], I) → B'_4
  104. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], H) → B'_5
  105. Vector(B, B'_4) → v_2
  106. Vector(B, B'_5) → w_2
  107. Line(I, p) → g_3
  108. Point(g_3) → N_1
  109. Line(H, n''') → l_3
  110. Point(l_3) → O_1
  111. Polygon(A, F, B) → poly3, b_2, a_2, f_3
  112. Circle(A, 2) → c_4
  113. Circle(F, 2) → d_4
  114. Polygon(F, A, 4) → poly4, i_4, j_4, k_4, l_4, C_2, D_2
  115. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], C_2) → A'_1
  116. Segment(A, A'_1) → m_2
  117. Rotate(A, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], C_2) → A'_2
  118. Rotate(A'_1, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], C_2) → J
  119. Segment(A'_2, J) → n_2
  120. Rotate(A'_2, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], D_2) → K
  121. Rotate(J, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], D_2) → J'
  122. Segment(K, J') → p_2
  123. Rotate(K, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], F) → K'
  124. Rotate(J', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], F) → J''
  125. Segment(K', J'') → q_2
  126. Angle(B, F, D_2) → ε
  127. Rotate(c_1, If[12 < a ≤ 13, ((a - 12) * ε), ε], F) → r_2
  128. Midpoint(A, D_2) → G_1
  129. Dilate(A, If[12 < a ≤ 13, 13 - a, 0], G_1) → A'_3
  130. Segment(A, A'_3) → s_2
  131. Rotate(A, If[13 < a ≤ 14, ((a - 13) * π), π], G_1) → A'_4
  132. Rotate(A'_3, If[13 < a ≤ 14, ((a - 13) * π), π], G_1) → H_1
  133. Segment(A'_4, H_1) → t_2
  134. Line(G_1, B) → h_3
  135. Line(A, I) → i_3
  136. Line(F, H) → j_3
  137. Intersect(i_3, j_3) → I_1
  138. Line(H, I) → k_3
  139. Distance(H, I) → distanceHI
  140. Distance(B, G_1) → distanceBG1
  141. Dilate(G_1, If[14 < a ≤ 15, 15 - a, 0], B) → J_1
  142. Segment(G_1, J_1) → m_3
  143. Dilate(A, If[15 < a ≤ 16, 16 - a, 0], I) → A'_5
  144. Segment(A, A'_5) → n_3
  145. Dilate(F, If[16 < a ≤ 17, 17 - a, 0], I_1) → F'_5
  146. Segment(F, F'_5) → p_3
  147. Dilate(I_1, If[17 < a ≤ 18, 18 - a, 0], H) → K_1
  148. Segment(I_1, K_1) → q_3
  149. Line(H, I) → r_3
  150. Line(B, G_1) → s_3
  151. Intersect(r_3, s_3) → L_1
  152. OrthogonalLine(C, r_3) → t_3
  153. Intersect(t_3, r_3) → M_1
  154. Line(H, I) → a_3
  155. Line(G_1, a_3) → b_3
  156. Rotate(G_1, If[18 < a ≤ 19, ((-(a - 18)) * π / 2), ((-π)) / 2], G_1) → P_1
  157. Rotate(J_1, If[18 < a ≤ 19, ((-(a - 18)) * π / 2), ((-π)) / 2], G_1) → Q_1
  158. Segment(P_1, Q_1) → c_3
  159. Dilate(G_1, If[19 < a ≤ 20, 20 - a, 0], H) → R_1
  160. Vector(G_1, R_1) → u_3
  161. Translate(P_1, u_3) → S_1
  162. Translate(Q_1, u_3) → T_1
  163. Segment(S_1, T_1) → d_3
  164. Dilate(C, If[20 < a ≤ 21, 21 - a, 0], M_1) → C'_2
  165. Vector(C, C'_2) → v_3
  166. Translate(e_1, v_3) → e_3
  167. Line(C, d_3) → f_4
  168. Point(f_4) → U_1
  169. Angle(G, C, U_1) → ζ
  170. Rotate(e_3, If[21 < a ≤ 22, ((a - 21) * ζ), ζ], M_1) → e_4
  171. Dilate(M_1, If[22 < a ≤ 23, 23 - a, 0], L_1) → V_1
  172. Vector(M_1, V_1) → w_3
  173. Translate(e_4, w_3) → g_4
  174. Line(I_1, B) → h_4
  175. Circle(B, 1) → p_4
  176. Intersect(p_4, h_4, 2) → W_1
  177. Ray(I_1, B) → m_4
  178. Circle(G_1, 2) → q_4
  179. Intersect(q_4, s_3, 2) → Z_1
  180. Ray(G_1, Z_1) → n_4