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Dynamic Geometry Illustration by Tim Brzezinski (9)

Dynamic Geometry Illustration by Tim Brzezinski (9)
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2017-12-17 09:25
Solid GeometryTransformationGeometric Construction
This example dynamically demonstrates polygon construction and related geometric transformations. It involves core concepts like area, centroid, dilation, quadrilaterals, limits, solids, random experiments, and straight lines. Construction highlights include commands such as Polygon, Midpoint, Segment, Intersect, Angle, Circle, Min, Dilate, Rotate, Vector, Translate, and CircleSector. Suitable for high school students and math enthusiasts exploring geometric transformations and limit phenomena.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → q1, a, b, c, d
  2. Midpoint(c) → E
  3. Midpoint(b) → F
  4. Midpoint(a) → G
  5. Midpoint(d) → H
  6. Segment(A, C) → f
  7. Segment(D, B) → g
  8. Segment(C, H) → h
  9. Segment(A, E) → i
  10. Intersect(i, h) → I
  11. Segment(B, E) → j
  12. Segment(D, F) → k
  13. Intersect(k, j) → J
  14. Segment(C, G) → l
  15. Segment(A, F) → m
  16. Intersect(m, l) → K
  17. Segment(B, H) → n
  18. Segment(D, G) → p
  19. Intersect(p, n) → L
  20. Polygon(I, J, K, L) → q2, i_1, j_1, k_1, l_1
  21. Angle(q2) → α, β, γ, δ
  22. Angle(q1) → ε, ζ, η, θ
  23. Circle(A, a / 3) → e
  24. Intersect(e, a, 1) → M
  25. Circle(M, a / 3) → q
  26. Intersect(q, a, 2) → N
  27. Circle(B, b / 3) → r
  28. Intersect(r, b, 1) → O
  29. Circle(O, b / 3) → s
  30. Intersect(s, b, 2) → P
  31. Circle(C, c / 3) → t
  32. Intersect(t, c, 1) → Q
  33. Circle(Q, c / 3) → c_1
  34. Intersect(c_1, c, 2) → R
  35. Circle(D, d / 3) → d_1
  36. Intersect(d_1, d, 1) → S
  37. Circle(S, d / 3) → e_1
  38. Intersect(e_1, d, 2) → T
  39. Min({l_1, i_1, j_1, k_1}) → o
  40. Circle(L, (0.35 * o)) → g_1
  41. Circle(K, (0.35 * o)) → p_1
  42. Circle(J, (0.35 * o)) → h_1
  43. Circle(I, (0.35 * o)) → f_1
  44. Segment(A, D) → m_1
  45. Segment(D, C) → n_1
  46. Segment(C, B) → q_1
  47. Segment(B, A) → r_1
  48. Segment(L, K) → s_1
  49. Segment(K, J) → t_1
  50. Segment(J, I) → a_1
  51. Segment(I, L) → b_1
  52. Intersect(f_1, b_1, 1) → U
  53. Intersect(g_1, s_1, 1) → V
  54. Intersect(p_1, t_1, 1) → W
  55. Intersect(h_1, a_1) → Z,
  56. Dilate(A, If[0 < u ≤ 1, 1 - u, 0], H) → A'
  57. Dilate(D, If[0 < u ≤ 1, 1 - u, 0], E) → D'
  58. Segment(A, A') → f_2
  59. Segment(D, D') → g_2
  60. Rotate(A, If[1 < u ≤ 2, ((u - 1) * π), π], H) → A'_1
  61. Rotate(A', If[1 < u ≤ 2, ((u - 1) * π), π], H) → A''
  62. Segment(A'_1, A'') → h_2
  63. Rotate(D, If[1 < u ≤ 2, ((u - 1) * π), π], E) → D'_1
  64. Rotate(D', If[1 < u ≤ 2, ((u - 1) * π), π], E) → D''
  65. Segment(D'_1, D'') → i_2
  66. Dilate(A, If[2 < u ≤ 3, 3 - u, 0], E) → A'_2
  67. Dilate(C, If[2 < u ≤ 3, 3 - u, 0], H) → C'
  68. Segment(A, A'_2) → j_2
  69. Segment(C, C') → k_2
  70. Dilate(D, If[6 < u ≤ 7, 7 - u, 0], F) → D'_2
  71. Dilate(B, If[6 < u ≤ 7, 7 - u, 0], E) → B'
  72. Segment(D, D'_2) → l_2
  73. Segment(B, B') → m_2
  74. Dilate(C, If[4 < u ≤ 5, 5 - u, 0], F) → C'_1
  75. Dilate(B, If[4 < u ≤ 5, 5 - u, 0], G) → B'_1
  76. Segment(C, C'_1) → n_2
  77. Segment(B, B'_1) → p_2
  78. Rotate(B, If[5 < u ≤ 6, ((u - 5) * π), π], G) → B'_2
  79. Rotate(B'_1, If[5 < u ≤ 6, ((u - 5) * π), π], G) → A_1
  80. Segment(B'_2, A_1) → q_2
  81. Rotate(C, If[5 < u ≤ 6, ((u - 5) * π), π], F) → C'_2
  82. Rotate(C'_1, If[5 < u ≤ 6, ((u - 5) * π), π], F) → B_1
  83. Segment(C'_2, B_1) → r_2
  84. Dilate(C, If[8 < u ≤ 9, 9 - u, 0], G) → F'
  85. Dilate(A, If[8 < u ≤ 9, 9 - u, 0], F) → A'_3
  86. Segment(C, F') → s_2
  87. Segment(A, A'_3) → t_2
  88. Dilate(B, If[10 < u ≤ 11, 11 - u, 0], H) → B'_3
  89. Dilate(D, If[10 < u ≤ 11, 11 - u, 0], G) → D'_3
  90. Segment(D, D'_3) → a_2
  91. Segment(B, B'_3) → b_2
  92. Dilate(I, If[12 < u ≤ 13, 13 - u, 0], J) → I'
  93. Segment(I, I') → c_2
  94. Dilate(I, If[13 < u ≤ 14, 14 - u, 0], N) → I'_1
  95. Vector(I, I'_1) → v
  96. Translate(I, v) → I'_2
  97. Translate(I', v) → I''
  98. Segment(I'_2, I'') → d_2
  99. Dilate(I, If[14 < u ≤ 15, 15 - u, 0], M) → I'_3
  100. Vector(I, I'_3) → w
  101. Translate(I, w) → I'_4
  102. Translate(I', w) → I''_1
  103. Segment(I'_4, I''_1) → e_2
  104. Dilate(I, If[15 < u ≤ 16, 16 - u, 0], A) → I'_5
  105. Vector(I, I'_5) → u_1
  106. Translate(I, u_1) → I'_6
  107. Translate(I', u_1) → I''_2
  108. Segment(I'_6, I''_2) → f_3
  109. Dilate(J, If[16 < u ≤ 17, 17 - u, 0], K) → J'
  110. Segment(J, J') → g_3
  111. Dilate(J, If[17 < u ≤ 18, 18 - u, 0], D) → J'_1
  112. Dilate(J, If[18 < u ≤ 19, 19 - u, 0], S) → J'_2
  113. Dilate(J, If[19 < u ≤ 20, 20 - u, 0], T) → J'_3
  114. Vector(J, J'_1) → v_1
  115. Vector(J, J'_2) → w_1
  116. Vector(J, J'_3) → u_2
  117. Translate(J, v_1) → J'_4
  118. Translate(J', v_1) → J''
  119. Segment(J'_4, J'') → h_3
  120. Translate(J, w_1) → J'_5
  121. Translate(J', w_1) → J''_1
  122. Segment(J'_5, J''_1) → i_3
  123. Translate(J, u_2) → J'_6
  124. Translate(J', u_2) → J''_2
  125. Segment(J'_6, J''_2) → j_3
  126. Dilate(K, If[20 < u ≤ 21, 21 - u, 0], L) → K'
  127. Segment(K, K') → k_3
  128. Dilate(L, If[21 < u ≤ 22, 22 - u, 0], D) → L'
  129. Dilate(L, If[22 < u ≤ 23, 23 - u, 0], R) → L'_1
  130. Dilate(L, If[23 < u ≤ 24, 24 - u, 0], Q) → L'_2
  131. Dilate(K, If[21 < u ≤ 22, 22 - u, 0], C) → K'_1
  132. Vector(K, K'_1) → v_2
  133. Translate(K, v_2) → K'_2
  134. Translate(K', v_2) → K''
  135. Segment(K'_2, K'') → l_3
  136. Dilate(K, If[22 < u ≤ 23, 23 - u, 0], Q) → K'_3
  137. Vector(K, K'_3) → w_2
  138. Translate(K, w_2) → K'_4
  139. Translate(K', w_2) → K''_1
  140. Segment(K'_4, K''_1) → m_3
  141. Dilate(K, If[23 < u ≤ 24, 24 - u, 0], R) → L'_3
  142. Vector(K, L'_3) → u_3
  143. Translate(K, u_3) → K'_5
  144. Translate(K', u_3) → K''_2
  145. Segment(K'_5, K''_2) → n_3
  146. Dilate(L, If[24 < u ≤ 25, 25 - u, 0], I) → L'_4
  147. Segment(L, L'_4) → p_3
  148. Dilate(L, If[25 < u ≤ 26, 26 - u, 0], B) → L'_5
  149. Dilate(L, If[26 < u ≤ 27, 27 - u, 0], O) → L'_6
  150. Dilate(L, If[27 < u ≤ 28, 28 - u, 0], P) → L'_7
  151. Vector(L, L'_5) → v_3
  152. Vector(L, L'_6) → w_3
  153. Vector(L, L'_7) → a_3
  154. Translate(L, v_3) → L'_8
  155. Translate(L'_4, v_3) → C_1
  156. Segment(L'_8, C_1) → q_3
  157. Translate(L, w_3) → L'_9
  158. Translate(L'_4, w_3) → D_1
  159. Segment(L'_9, D_1) → r_3
  160. Translate(L, a_3) → L'_{10}
  161. Translate(L'_4, a_3) → E_1
  162. Segment(L'_{10}, E_1) → s_3
  163. Rotate(V, If[30 < u ≤ 31, ((u - 30) * δ), δ], L) → V'
  164. CircleSector(L, V, V') → c_3
  165. Rotate(W, If[31 < u ≤ 32, ((u - 31) * γ), γ], K) → W'
  166. CircleSector(K, W, W') → d_3
  167. Rotate(Z, If[32 < u ≤ 33, ((u - 32) * β), β], J) → Z'
  168. CircleSector(J, Z, Z') → e_3
  169. Dilate(L, If[33 < u ≤ 34, 34 - u, 0], C) → L'_{11}
  170. Vector(L, L'_{11}) → b_3
  171. Translate(c_3, b_3) → t_3
  172. Rotate(t_3, If[34 < u ≤ 35, ((u - 34) * π), π], C) → c_4
  173. Dilate(K, If[35 < u ≤ 36, 36 - u, 0], D) → K'_6
  174. Vector(K, K'_6) → u_4
  175. Translate(d_3, u_4) → d_4
  176. Rotate(d_4, If[36 < u ≤ 37, ((u - 36) * π), π], D) → e_4
  177. Dilate(J, If[37 < u ≤ 38, 38 - u, 0], A) → J'_7
  178. Vector(J, J'_7) → v_4
  179. Translate(e_3, v_4) → f_4
  180. Rotate(f_4, If[38 < u ≤ 39, ((u - 38) * π), π], A) → g_4

AI Drawing Prompt Reference

Create a dynamic geometry illustration showing polygon construction and related geometric transformations. Start by drawing a quadrilateral and find the midpoints of its sides, connecting them to form an inner quadrilateral. Then calculate the midpoints of the diagonals and connect these points to create a new shape. Next, draw circles centered at the vertices of the quadrilateral with radii proportional to the sides, and find their intersections with the sides. Finally, draw smaller circles at these intersection points and connect certain points with segments to illustrate concepts like area, centroid, dilation, rotation, and straight lines.
The prompt above was compiled from the original drawing commands and is for reference only.