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Square Minus Kite

Square Minus Kite
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2017-05-02 01:00
PolygonTransformationGeometric Construction
This example demonstrates the result of subtracting a kite from a square. Key mathematical concepts include geometric transformations and polygons. Using GeoGebra commands such as Ray, Segment, Circle, Intersect, Rotate, Dilate, etc., the figure can be dynamically modified. Suitable for students in middle school and above to learn about geometric transformations and polygon-related knowledge.

GeoGebra Commands & Parameters

  1. Ray(A, B_2) → m_2
  2. Segment(A, B_2) → g
  3. Circle(B_2, g) → q_2
  4. Intersect(q_2, m_2, 2) → J_1
  5. Segment(B_2, J_1) → n_2
  6. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  7. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  8. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  9. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  10. Midpoint(A, C) → V
  11. CircleArc(V, C, D) → g_2
  12. Point(g_2) → B
  13. Segment(C, B) → j_1
  14. Segment(A, B) → n
  15. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  16. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  17. Segment(R, A''') → f''
  18. Intersect(n, f'') → Z
  19. Segment(B, Z) → l_1
  20. Segment(A, A') → f
  21. Segment(A'_1, A'') → f'
  22. Segment(B_2, C) → h
  23. Segment(C, D) → i
  24. Circle(B_2, (0.1 * Distance[A, B_2])) → p
  25. Segment(D, A) → j
  26. Circle(C, Radius[p]) → k
  27. Circle(A, (0.1 * j)) → q
  28. Circle(D, (0.1 * j)) → r
  29. Segment(A, C) → l
  30. Segment(D, B_2) → m
  31. Intersect(r, j, 1) → F
  32. Intersect(r, m, 1) → G
  33. Intersect(r, i, 1) → H
  34. Intersect(k, i, 1) → I
  35. Intersect(k, l, 1) → J
  36. Intersect(k, h, 1) → K
  37. Intersect(p, h, 1) → L
  38. Intersect(p, m, 1) → M
  39. Intersect(p, g, 1) → N
  40. Intersect(q, g, 1) → O
  41. Intersect(q, l, 1) → P
  42. Intersect(q, j, 1) → Q
  43. CircleSector(B_2, L, N) → s
  44. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  45. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  46. Segment(R', S) → f'''
  47. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  48. Vector(B_2, B') → u
  49. Translate(s, u) → s'
  50. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  51. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  52. Vector(C, C') → v
  53. Translate(s'', v) → s'''
  54. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  55. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  56. Vector(D, D') → w
  57. Translate(t, w) → t'
  58. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  59. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  60. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  61. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  62. Vector(B_2, B'_1) → v_1
  63. Vector(B_2, B'_3) → w_1
  64. Vector(B_2, B'_2) → i_1
  65. Circle(A, n - j_1) → h_2
  66. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  67. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  68. Vector(B_2, B'_4) → u_1
  69. Translate(s, u_1) → s'_1
  70. Line(B, f') → a_1
  71. Point(a_1) → U
  72. Angle(U, B, C) → α
  73. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  74. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  75. Line(A, B_2) → h_1
  76. Angle(B, A, D) → β
  77. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  78. Mirror(C, D) → C'_1
  79. Rotate(C, ((-π)) / 3, B_2) → C'_2
  80. Segment(B_2, C'_2) → f_1
  81. Segment(C, C'_2) → g_1
  82. Intersect(k, g_1, 1) → E
  83. Intersect(k, h) → T,
  84. CircleSector(C, T, E) → d_1
  85. Rotate(D, 15°, C) → D'_2
  86. Angle(D, C, D'_2) → γ
  87. Ray(C, D'_2) → k_1
  88. Rotate(C, (-(15°)), D) → C'_3
  89. Angle(C'_3, D, C) → δ
  90. Ray(D, C'_3) → m_1
  91. Intersect(k_1, m_1) → W
  92. Rotate(B_2, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → A_1
  93. Rotate(C, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → C'_4
  94. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  95. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  96. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  97. Polygon(C, W, D) → poly2, d_2, c_2, w_2
  98. Polygon(A, W, D) → poly3, d_3, a_2, w_3
  99. Polygon(B_2, W, C) → poly4, c_3, b_2, w_4
  100. Polygon(A, B_2, W) → poly5, w_5, a_3, b_1
  101. Rotate(T, 15°, C) → T'
  102. Angle(T, C, T') → ε
  103. Ray(C, T') → q_1
  104. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  105. Vector(C, C'_8) → u_2
  106. Dilate(D, If[23 < e ≤ 24, 24 - e, 0], W) → D'_3
  107. Dilate(C, If[23 < e ≤ 24, 24 - e, 0], W) → C'_9
  108. Dilate(B_2, If[25 < e ≤ 26, 26 - e, 0], W) → G_1
  109. Dilate(A, If[25 < e ≤ 26, 26 - e, 0], W) → A'_2
  110. Segment(A, A'_2) → j_2
  111. Segment(B_2, G_1) → l_2
  112. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  113. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  114. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  115. Polygon(H_1, I_1, C'_{10}) → poly6', c'_{10}, h_3, i_3
  116. Polygon(A, B_2, C, D) → poly7, a_4, b_4, c_4, d_4
  117. Segment(B_2, C) → j_3
  118. Point(j_3) → V_1
  119. Ray(D, V_1) → k_3
  120. Ray(A, B_2) → n_3
  121. Intersect(k_3, n_3) → W_1
  122. Angle(V_1, D, C) → κ
  123. Angle(C, V_1, D) → θ
  124. Segment(V_1, C) → p_3
  125. Rotate(V_1, (2 * θ), V_1) → Z_1
  126. Rotate(C, (2 * θ), V_1) → C'_7
  127. Segment(Z_1, C'_7) → q_3
  128. Ray(D, C'_7) → r_3
  129. Intersect(r_3, n_3) → A_2
  130. Segment(A, A_2) → s_3
  131. Segment(A_2, W_1) → t_3
  132. Polygon(C'_7, V_1, C, D) → poly8, c'_7, v_3, c_5, d_5
  133. Polygon(D, A, B_2, V_1, C'_7) → poly9, d_6, a_5, b_5, v_4, c'_1
  134. Dilate(C, If[10 < e ≤ 11, 11 - e, 0], V_1) → C'_{11}
  135. Segment(C, C'_{11}) → n_1
  136. Dilate(A, If[14 < e ≤ 15, 15 - e, 0], A_2) → A'_4
  137. Segment(A, A'_4) → p_1
  138. Rotate(C, If[11 < e ≤ 12, (((e - 11) * 2) * θ), (2 * θ)], V_1) → C'_{12}
  139. Rotate(C'_{11}, If[11 < e ≤ 12, (((e - 11) * 2) * θ), (2 * θ)], V_1) → U_1
  140. Segment(C'_{12}, U_1) → r_1
  141. Rotate(R, If[12 < e ≤ 13, (((-(e - 12)) * 2) * κ), (-2 * κ)], D) → R'_1
  142. Rotate(A''', If[12 < e ≤ 13, (((-(e - 12)) * 2) * κ), (-2 * κ)], D) → C_2
  143. Segment(R'_1, C_2) → f'''_1
  144. Polygon(D, C, V_1, C'_7) → poly10, d_7, c_6, v_5, c'_2
  145. Dilate(C'_{12}, If[13 < e ≤ 14, 14 - e, 0], A_2) → D_2
  146. Segment(C'_{12}, D_2) → s_1
  147. Dilate(D, If[15 < e ≤ 16, 16 - e, 0], V_1) → D'_4
  148. Segment(D, D'_4) → t_1
  149. Dilate(V_1, If[16 < e ≤ 17, 17 - e, 0], W_1) → C_1
  150. Segment(V_1, C_1) → c_1
  151. Line(A, B_2) → e_1
  152. Circle(A_2, p_1) → e_2
  153. Intersect(e_2, e_1, 2) → D_1
  154. Dilate(A_2, If[15 < e ≤ 16, 16 - e, 0], D_1) → E_1
  155. Segment(A_2, E_1) → f_2
  156. Dilate(D_1, If[16 < e ≤ 17, 17 - e, 0], W_1) → F_1
  157. Segment(D_1, F_1) → i_2
  158. Rotate(A, If[17 < e ≤ 18, ((-(e - 17)) * π), (-π)], A_2) → A'_3
  159. Rotate(A'_4, If[17 < e ≤ 18, ((-(e - 17)) * π), (-π)], A_2) → K_1
  160. Segment(A'_3, K_1) → k_2
  161. Dilate(V_1, If[18 < e ≤ 19, 19 - e, 0], D_1) → L_1
  162. Vector(V_1, L_1) → v_2
  163. Translate(C, v_2) → C'_6
  164. Translate(C'_{11}, v_2) → M_1
  165. Segment(C'_6, M_1) → p_2
  166. Rotate(C'_6, If[19 < e ≤ 20, ((-(e - 19)) * π / 2), ((-π)) / 2], D_1) → N_1
  167. Rotate(M_1, If[19 < e ≤ 20, ((-(e - 19)) * π / 2), ((-π)) / 2], D_1) → O_1
  168. Segment(N_1, O_1) → r_2

AI Drawing Prompt Reference

Draw a square and a kite to demonstrate the result of subtracting the kite from the square. Start by sketching the sides and vertices of the square, then define the vertices of the kite by drawing its diagonals. Apply geometric transformations such as rotation and dilation to the kite, then subtract it from the square. Finally, add some guide lines and arcs to highlight the geometric transformations and key points, completing the entire figure.
The prompt above was compiled from the original drawing commands and is for reference only.