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Dynamic Demonstration of Rhombus Rotation Phenomenon

Dynamic Demonstration of Rhombus Rotation Phenomenon
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2018-09-15 08:23
PolygonGeometric Construction
This is a dynamic demonstration of a geometry challenge proof problem, showcasing an interesting phenomenon of a rhombus during rotation. Key mathematical concepts include the properties of a rhombus and rotation transformations. It is constructed using GeoGebra commands such as Segment, Rotate, Angle, Polygon, etc. Suitable for middle school and above students to learn about geometric transformations and proofs.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Rotate(B, (a * π / 180), A) → B'
  3. Angle(B, A, B') → α
  4. Ray(A, B') → g
  5. Point(g) → C
  6. Polygon(A, B, C) → t1, c, a_1, b
  7. Segment(A, C) → h
  8. Segment(C, B) → i
  9. Circle(A, f / 3) → d
  10. Intersect(d, f, 1) → D
  11. Midpoint(D, B) → E
  12. Circle(A, h / 3) → e
  13. Intersect(e, h, 1) → F
  14. Midpoint(F, C) → G
  15. Circle(C, i / 3) → k
  16. Intersect(k, i, 1) → H
  17. Midpoint(H, B) → I
  18. Segment(A, H) → j
  19. Segment(A, I) → l
  20. Segment(C, D) → m
  21. Segment(C, E) → n
  22. Segment(B, F) → p
  23. Segment(B, G) → q
  24. Intersect(j, q) → J
  25. Intersect(m, p) → K
  26. Intersect(l, n) → L
  27. Polygon(K, L, J) → t2, j_1, k_1, l_1
  28. Vector(K, A) → u
  29. Translate(K, u) → K'
  30. Translate(L, u) → L'
  31. Translate(J, u) → J'
  32. Polygon(K', L', J') → t2', j', k', l'
  33. Vector(K, L') → v
  34. Translate(K, v) → K'_1
  35. Translate(L, v) → L'_1
  36. Translate(J, v) → J'_1
  37. Polygon(K'_1, L'_1, J'_1) → t2'_1, j'_1, k'_1, l'_1
  38. Vector(L, E) → w
  39. Vector(J, C) → r
  40. Translate(K, r) → K'_2
  41. Translate(L, r) → L'_2
  42. Translate(J, r) → J'_2
  43. Polygon(K'_2, L'_2, J'_2) → t2'_2, j'_2, k'_2, l'_2
  44. Rotate(K, π, J) → K'_3
  45. Rotate(L, π, J) → L'_3
  46. Rotate(J, π, J) → J'_3
  47. Polygon(K'_3, L'_3, J'_3) → t2'_3, j'_3, k'_3, l'_3
  48. Vector(K, J') → t
  49. Translate(K, t) → K'_4
  50. Translate(L, t) → L'_4
  51. Translate(J, t) → J'_4
  52. Polygon(K'_4, L'_4, J'_4) → t2'_4, j'_4, k'_4, l'_4
  53. Circle(L'_1, j') → c_1
  54. Intersect(c_1, f, 2) → M
  55. Vector(L, M) → u_1
  56. Translate(K, u_1) → K'_5
  57. Translate(L, u_1) → L'_5
  58. Translate(J, u_1) → J'_5
  59. Polygon(K'_5, L'_5, J'_5) → t2'_5, j'_5, k'_5, l'_5
  60. Vector(L', L'_1) → v_1
  61. Translate(K'_5, v_1) → N
  62. Translate(L'_5, v_1) → O
  63. Translate(J'_5, v_1) → P
  64. Polygon(N, O, P) → t3, p_1, n_1, o_1
  65. Translate(K, v_1) → K'_6
  66. Translate(L, v_1) → L'_6
  67. Translate(J, v_1) → J'_6
  68. Polygon(K'_6, L'_6, J'_6) → t2'_6, j'_6, k'_6, l'_6
  69. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], D) → A'
  70. Segment(A, A') → f_1
  71. Rotate(A, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], D) → A'_1
  72. Segment(D, A'_1) → g_1
  73. Rotate(D, If[2 < o ≤ 3, ((-(o - 2)) * π), (-π)], E) → D'
  74. Segment(E, D') → h_1
  75. Dilate(B, If[3 < o ≤ 4, 4 - o, 0], I) → B'_1
  76. Segment(B, B'_1) → i_1
  77. Rotate(B, If[4 < o ≤ 5, ((-(o - 4)) * π), (-π)], I) → B'_2
  78. Segment(I, B'_2) → m_1
  79. Rotate(I, If[5 < o ≤ 6, ((-(o - 5)) * π), (-π)], H) → I'
  80. Segment(H, I') → q_1
  81. Dilate(C, If[6 < o ≤ 7, 7 - o, 0], G) → C'
  82. Segment(C, C') → r_1
  83. Rotate(C, If[7 < o ≤ 8, ((-(o - 7)) * π), (-π)], G) → C'_1
  84. Segment(G, C'_1) → s_1
  85. Rotate(G, If[8 < o ≤ 9, ((-(o - 8)) * π), (-π)], F) → G'
  86. Segment(F, G') → t_1
  87. Dilate(A, If[10 < o ≤ 11, 11 - o, 0], J) → A'_2
  88. Segment(A, A'_2) → b_1
  89. Dilate(B, If[10 < o ≤ 11, 11 - o, 0], J) → B'_3
  90. Segment(B, B'_3) → d_1
  91. Dilate(J, If[11 < o ≤ 12, 12 - o, 0], G) → J'_7
  92. Dilate(J, If[11 < o ≤ 12, 12 - o, 0], H) → J'_8
  93. Segment(J, J'_7) → e_1
  94. Segment(J, J'_8) → f_2
  95. Dilate(A, If[12 < o ≤ 13, 13 - o, 0], L) → A'_3
  96. Dilate(C, If[12 < o ≤ 13, 13 - o, 0], L) → C'_2
  97. Segment(C, C'_2) → g_2
  98. Segment(A, A'_3) → h_2
  99. Dilate(L, If[13 < o ≤ 14, 14 - o, 0], E) → L'_7
  100. Dilate(L, If[13 < o ≤ 14, 14 - o, 0], I) → L'_8
  101. Segment(L, L'_8) → i_2
  102. Segment(L, L'_7) → j_2
  103. Dilate(C, If[14 < o ≤ 15, 15 - o, 0], K) → C'_3
  104. Dilate(B, If[14 < o ≤ 15, 15 - o, 0], K) → B'_4
  105. Segment(C, C'_3) → k_2
  106. Segment(B, B'_4) → l_2
  107. Dilate(K, If[15 < o ≤ 16, 16 - o, 0], D) → K'_7
  108. Dilate(K, If[15 < o ≤ 16, 16 - o, 0], F) → K'_8
  109. Segment(K, K'_7) → m_2
  110. Segment(K, K'_8) → n_2
  111. Dilate(J, If[18 < o ≤ 19, 19 - o, 0], K) → J'_9
  112. Segment(J, J'_9) → p_2
  113. Dilate(K, If[18 < o ≤ 19, 19 - o, 0], L) → K'_9
  114. Segment(K, K'_9) → q_2
  115. Dilate(L, If[18 < o ≤ 19, 19 - o, 0], J) → L'_9
  116. Segment(L, L'_9) → r_2
  117. Dilate(K, If[23 < o ≤ 24, 24 - o, 0], A) → K'_{10}
  118. Vector(K, K'_{10}) → w_1
  119. Translate(K, w_1) → K'_{11}
  120. Translate(L, w_1) → L'_{10}
  121. Translate(J, w_1) → J'_{10}
  122. Polygon(K'_{11}, L'_{10}, J'_{10}) → t2'_7, j'_{10}, k'_{11}, l'_{10}
  123. Translate(J, w_1) → J'_{11}
  124. Translate(L, w_1) → L'_{11}
  125. Midpoint(A, B) → R
  126. Midpoint(A, C) → S
  127. Midpoint(C, B) → T
  128. Dilate(A, If[25 < o ≤ 26, 26 - o, 0], K'_1) → A'_4
  129. Dilate(A, If[25 < o ≤ 26, 26 - o, 0], R) → A'_5
  130. Dilate(A, If[25 < o ≤ 26, 26 - o, 0], N) → A'_6
  131. Vector(A, A'_4) → u_2
  132. Vector(A, A'_5) → v_2
  133. Vector(A, A'_6) → w_2
  134. Translate(K'_{11}, u_2) → Q
  135. Translate(L'_{10}, u_2) → U
  136. Translate(J'_{10}, u_2) → V
  137. Polygon(Q, U, V) → t4, v_3, q_3, u_3
  138. Translate(K'_{11}, w_2) → W
  139. Translate(L'_{10}, w_2) → Z
  140. Translate(J'_{10}, w_2) → A_1
  141. Polygon(W, Z, A_1) → t5, a_2, w_3, z_1
  142. Translate(K'_{11}, v_2) → B_1
  143. Translate(L'_{10}, v_2) → C_1
  144. Translate(J'_{10}, v_2) → D_1
  145. Polygon(B_1, C_1, D_1) → t6, d_2, b_2, c_2
  146. Dilate(K'_1, If[26 < o ≤ 27, 27 - o, 0], K) → E_1
  147. Vector(K'_1, E_1) → e_2
  148. Translate(Q, e_2) → Q'
  149. Translate(U, e_2) → U'
  150. Translate(V, e_2) → V'
  151. Polygon(Q', U', V') → t4', v', q', u'
  152. Translate(B_1, e_2) → F_1
  153. Translate(C_1, e_2) → G_1
  154. Translate(D_1, e_2) → H_1
  155. Polygon(F_1, G_1, H_1) → t6', h_3, f_3, g_3
  156. Translate(K'_{11}, Vector[(2 * e_2)]) → I_1
  157. Translate(L'_{10}, Vector[(2 * e_2)]) → J_1
  158. Translate(J'_{10}, Vector[(2 * e_2)]) → K_1
  159. Polygon(I_1, J_1, K_1) → t7, k_3, i_3, j_3
  160. Translate(K'_{11}, e_2) → L_1
  161. Translate(L'_{10}, e_2) → M_1
  162. Translate(J'_{10}, e_2) → N_1
  163. Polygon(L_1, M_1, N_1) → t8, n_3, l_3, m_3
  164. Translate(K'_{11}, Vector[(3 * e_2)]) → O_1
  165. Translate(L'_{10}, Vector[(3 * e_2)]) → P_1
  166. Translate(J'_{10}, Vector[(3 * e_2)]) → Q_1
  167. Polygon(O_1, P_1, Q_1) → t9, q_4, o_3, p_3
  168. Translate(Q, Vector[(2 * e_2)]) → Q'_1
  169. Translate(U, Vector[(2 * e_2)]) → U'_1
  170. Translate(V, Vector[(2 * e_2)]) → V'_1
  171. Polygon(Q'_1, U'_1, V'_1) → t4'_1, v'_1, q'_1, u'_1
  172. Midpoint(r_2) → R_1
  173. Rotate(Q', If[28 < o ≤ 29, ((o - 28) * π), π], R_1) → Q''
  174. Rotate(U', If[28 < o ≤ 29, ((o - 28) * π), π], R_1) → U''
  175. Rotate(V', If[28 < o ≤ 29, ((o - 28) * π), π], R_1) → V''
  176. Polygon(Q'', U'', V'') → t4'', v'', q'', u''
  177. Vector(L, N) → s_2
  178. Translate(Q'', s_2) → Q'''
  179. Translate(U'', s_2) → U'''
  180. Translate(V'', s_2) → V'''
  181. Polygon(Q''', U''', V''') → t4''', v''', q''', u'''
  182. Translate(Q'', Vector[(-s_2)]) → Q'''_1
  183. Translate(U'', Vector[(-s_2)]) → U'''_1
  184. Translate(V'', Vector[(-s_2)]) → V'''_1
  185. Polygon(Q'''_1, U'''_1, V'''_1) → t4'''_1, v'''_1, q'''_1, u'''_1
  186. Vector(L, K) → t_2
  187. Translate(Q'', t_2) → Q'''_2
  188. Translate(U'', t_2) → U'''_2
  189. Translate(V'', t_2) → V'''_2
  190. Polygon(Q'''_2, U'''_2, V'''_2) → t4'''_2, v'''_2, q'''_2, u'''_2
  191. Translate(Q''', t_2) → S_1
  192. Translate(U''', t_2) → T_1
  193. Translate(V''', t_2) → U_1
  194. Polygon(S_1, T_1, U_1) → t10, u_4, s_3, t_3
  195. Translate(S_1, t_2) → V_1
  196. Translate(T_1, t_2) → W_1
  197. Translate(U_1, t_2) → Z_1
  198. Polygon(V_1, W_1, Z_1) → t10', z_2, v_4, w_4
  199. Polygon(J, K, L) → t11, l_4, j_4, k_4

AI Drawing Prompt Reference

Draw a rhombus and rotate it around one vertex to observe its dynamic changes. First, create a rhombus, then rotate it around one vertex to see the transformation. The process includes creating segments, rotating points, constructing angles, drawing polygons and circles, and finally using translation and rotation to demonstrate the rotation of the rhombus.
The prompt above was compiled from the original drawing commands and is for reference only.