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Dynamic Geometry Illustration of a Theorem with Square, Circles, and Tangent Cir

Dynamic Geometry Illustration of a Theorem with Square, Circles, and Tangent Cir
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2017-03-18 10:51
CirclePolygonVectorGeometric Construction
This example dynamically illustrates a theorem involving a square, two circles, and a mutually tangent circle. Key math concepts include circles, squares, geometric mean, vectors, and geometric transformations. Construction highlights the use of commands like Circle, Rotate, Dilate, Segment, Polygon, Point, OrthogonalLine, Intersect, Line, Radius, CircleSector, Vector, Translate, CircleArc, and Midpoint. Suitable for high school mathematics students and teachers to explore geometric theorems and transformations.

GeoGebra Commands & Parameters

  1. Circle(A, B) → e_1
  2. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B) → A'_1
  3. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B) → A'
  4. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B) → A''
  5. Segment(A, A') → f
  6. Circle(A, Radius[e_1] - f / 6) → p_1
  7. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  8. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  9. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  10. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  11. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  12. Segment(R', S) → f'''
  13. Point(f''') → Z
  14. Circle(Z, f / 6) → k_1
  15. OrthogonalLine(Z, f''') → g_1
  16. Intersect(k_1, g_1, 1) → A_1
  17. Point(f''') → W
  18. Circle(W, f / 6) → h_1
  19. OrthogonalLine(W, f''') → f_1
  20. Intersect(h_1, f_1, 1) → B_1
  21. Line(A_1, B_1) → j_1
  22. Intersect(p_1, j_1, 1) → V
  23. Circle(V, f / 6) → q_1
  24. Radius(q_1) → n
  25. Segment(A'_1, A'') → f'
  26. Segment(A, B) → g
  27. Segment(B, C) → h
  28. Segment(C, D) → i
  29. Segment(D, A) → j
  30. Circle(B, (0.35 * n)) → p
  31. Circle(C, (0.15 * j)) → k
  32. Circle(A, (0.15 * j)) → q
  33. Circle(D, (0.15 * j)) → r
  34. Segment(A, C) → l
  35. Segment(D, B) → m
  36. Intersect(l, m) → E
  37. Intersect(r, j, 1) → F
  38. Intersect(r, m, 1) → G
  39. Intersect(r, i, 1) → H
  40. Intersect(k, i, 1) → I
  41. Intersect(k, l, 1) → J
  42. Intersect(k, h, 1) → K
  43. Intersect(p, h, 1) → L
  44. Intersect(p, m, 1) → M
  45. Intersect(p, g, 1) → N
  46. Intersect(q, g, 1) → O
  47. Intersect(q, l, 1) → P
  48. Intersect(q, j, 1) → Q
  49. Segment(R, A''') → f''
  50. CircleSector(B, L, N) → s
  51. Dilate(B, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  52. Vector(B, B') → u
  53. Translate(s, u) → s'
  54. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  55. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  56. Vector(C, C') → v
  57. Translate(s'', v) → s'''
  58. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  59. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  60. Vector(D, D') → w
  61. Translate(t, w) → t'
  62. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  63. Dilate(A, If[10 < e ≤ 11, 11 - e, 0], E) → A'_2
  64. Dilate(E, If[11 < e ≤ 12, 12 - e, 0], C) → E'
  65. Dilate(D, If[12 < e ≤ 13, 13 - e, 0], E) → D'_1
  66. Dilate(E, If[13 < e ≤ 14, 14 - e, 0], B) → E'_1
  67. Dilate(D, If[17 < e ≤ 18, 18 - e, 0], E) → D'_2
  68. Vector(D, D'_2) → u_1
  69. Rotate(E, If[15 < e ≤ 16, ((e - 15) * π / 2), 1.5707963267948966], E) → E'_2
  70. Rotate(E'_1, If[14 < e ≤ 15, ((e - 14) * π / 2), 1.5708], E) → T
  71. Rotate(E'_2, If[16 < e ≤ 17, ((e - 16) * π / 2), 1.5707963267948966], E) → U
  72. Rotate(T, If[15 < e ≤ 16, ((e - 15) * π / 2), 1.5708], E) → T'
  73. Rotate(U, If[17 < e ≤ 18, ((e - 17) * π / 2), 1.5707963267948966], E) → U'
  74. Rotate(T', If[16 < e ≤ 17, ((e - 16) * π / 2), 1.5708], E) → T''
  75. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  76. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  77. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  78. Vector(B, B'_1) → v_1
  79. Vector(B, B'_3) → w_1
  80. Vector(B, B'_2) → i_1
  81. Circle(B, C) → c_1
  82. Intersect(e_1, j_1, 1) → C_1
  83. Intersect(c_1, q_1) → D_1, E_1, ,
  84. Rotate(A, If[10 < e ≤ 11, ((-(e - 10)) * π / 2), ((-π)) / 2], B) → A'_3
  85. Rotate(A, If[10 < e ≤ 11, ((-(e - 10)) * π / 2), ((-π)) / 2], B) → A'_4
  86. CircleArc(B, A'_4, A) → r_1
  87. Rotate(B, If[11 < e ≤ 12, ((e - 11) * π / 2), π / 2], A) → B'_4
  88. CircleArc(A, B, B'_4) → s_1
  89. Rotate(D_1, If[12 < e ≤ 14, ((e - 12) * π), (2 * π)], V) → F_1
  90. CircleArc(V, D_1, F_1) → d_1
  91. OrthogonalLine(V, f''') → l_1
  92. Intersect(l_1, f''') → G_1
  93. Dilate(V, If[14 < e ≤ 15, 15 - e, 0], G_1) → V'
  94. Segment(V, V') → m_1
  95. Dilate(D, If[15 < e ≤ 16, 16 - e, 0], G_1) → D'_3
  96. Vector(D, D'_3) → a_1
  97. Translate(t, a_1) → t'_1
  98. Segment(A, B) → n_1
  99. Circle(A, 1.6411491042356954) → t_1
  100. Circle(A, m_1) → c_2
  101. Intersect(c_2, n_1, 1) → H_1
  102. Circle(H_1, m_1) → d_2
  103. Intersect(d_2, n_1, 2) → I_1
  104. Circle(I_1, m_1) → e_2
  105. Intersect(e_2, n_1, 2) → J_1
  106. Circle(J_1, m_1) → f_2
  107. Intersect(f_2, n_1, 2) → K_1
  108. Circle(K_1, m_1) → g_2
  109. Intersect(g_2, n_1, 2) → L_1
  110. Dilate(G_1, If[16 < e ≤ 17, 17 - e, 0], L_1) → M_1
  111. Vector(G_1, M_1) → b_1
  112. Translate(V, b_1) → V'_1
  113. Translate(V', b_1) → V''
  114. Segment(V'_1, V'') → h_2
  115. Dilate(G_1, If[17 < e ≤ 18, 18 - e, 0], K_1) → N_1
  116. Vector(G_1, N_1) → u_2
  117. Translate(V, u_2) → V'_2
  118. Translate(V', u_2) → V''_1
  119. Segment(V'_2, V''_1) → i_2
  120. Dilate(G_1, If[18 < e ≤ 19, 19 - e, 0], J_1) → P_1
  121. Vector(G_1, P_1) → v_2
  122. Translate(V, v_2) → V'_3
  123. Translate(V', v_2) → V''_2
  124. Segment(V'_3, V''_2) → j_2
  125. Dilate(G_1, If[19 < e ≤ 20, 20 - e, 0], I_1) → Q_1
  126. Vector(G_1, Q_1) → w_2
  127. Translate(V, w_2) → V'_4
  128. Translate(V', w_2) → V''_3
  129. Segment(V'_4, V''_3) → k_2
  130. Dilate(G_1, If[20 < e ≤ 21, 21 - e, 0], A) → R_1
  131. Vector(G_1, R_1) → a_2
  132. Translate(V, a_2) → V'_5
  133. Translate(V', a_2) → V''_4
  134. Segment(V'_5, V''_4) → l_2
  135. Midpoint(G_1, I_1) → O_1
  136. Dilate(G_1, If[21 < e ≤ 22, 22 - e, 0], O_1) → S_1
  137. Vector(G_1, S_1) → b_2
  138. Translate(V, b_2) → V'_6
  139. Translate(V', b_2) → V''_5
  140. Segment(V'_6, V''_5) → m_2
  141. Rotate(V'_6, If[22 < e ≤ 23, (((e - 22) * 3) * π / 4), (3 * π / 4)], V'_6) → T_1
  142. Rotate(V''_5, If[22 < e ≤ 23, (((e - 22) * 3) * π / 4), (3 * π / 4)], V'_6) → U_1
  143. Segment(T_1, U_1) → n_2
  144. Dilate(U_1, If[23 < e ≤ 24, 24 - e, 0], I_1) → V_1
  145. Vector(U_1, V_1) → p_2
  146. Translate(T_1, p_2) → W_1
  147. Translate(U_1, p_2) → Z_1
  148. Segment(W_1, Z_1) → q_2
  149. Rotate(W_1, If[24 < e ≤ 25, ((e - 24) * π / 4), π / 4], I_1) → A_2
  150. Rotate(Z_1, If[25 < e ≤ 26, ((e - 25) * π / 4), π / 4], I_1) → B_2
  151. Segment(A_2, B_2) → r_2