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Problem in a Parallelogram

Problem in a Parallelogram
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2017-09-13 18:54
PolygonTransformationGeometric Construction
This is a dynamic and modifiable illustration of a problem within a parallelogram. Using transformations like rotation and translation, various geometric relationships within the parallelogram can be explored. Commands used include Line, Ray, Rotate, Point, Midpoint, Intersect, Segment, Angle, Min, Circle, Dilate, CircleSector, Vector, Translate, Polygon, LineBisector, Distance. Suitable for middle school and above students to learn about geometric transformations and explore the properties of parallelograms.

GeoGebra Commands & Parameters

  1. Line(A, B) → s_1
  2. Ray(A, B) → f
  3. Rotate(B, a°, A) → B'
  4. Ray(A, B') → g
  5. Point(g) → C
  6. Midpoint(B, C) → W
  7. Line(W, s_1) → t_1
  8. Line(C, f) → i
  9. Line(B, g) → h
  10. Intersect(h, i) → D
  11. Segment(D, B) → k_1
  12. Intersect(t_1, k_1) → F_1
  13. Segment(D, F_1) → k_2
  14. Angle(B, A, B') → α
  15. Segment(C, D) → j
  16. Segment(A, B) → k
  17. Segment(B, D) → l
  18. Segment(A, C) → m
  19. Min(m, k) → b
  20. Circle(C, (0.2 * b)) → d
  21. Circle(A, (0.15 * b)) → e
  22. Circle(B, (0.2 * b)) → p
  23. Circle(D, (0.2 * b)) → c
  24. Intersect(e, k, 1) → E
  25. Intersect(e, m, 1) → F
  26. Intersect(p, k, 1) → G
  27. Intersect(p, l, 1) → H
  28. Intersect(d, g, 2) → I
  29. Ray(C, I) → q
  30. Intersect(c, h, 2) → J
  31. Ray(D, J) → r
  32. Intersect(c, i, 2) → K
  33. Ray(D, K) → s
  34. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], B) → A'
  35. Segment(A, A') → t
  36. Rotate(E, If[1 < o ≤ 2, ((o - 1) * α), α], A) → E'
  37. CircleSector(A, E, E') → c_1
  38. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], C) → A'_1
  39. Vector(A, A'_1) → u
  40. Translate(c_1, u) → d_1
  41. Segment(A, A'_1) → f_1
  42. Dilate(C, If[3 < o ≤ 4, 4 - o, 0], D) → C'
  43. Vector(C, C') → v
  44. Translate(d_1, v) → e_1
  45. Segment(C, C') → g_1
  46. Dilate(D, If[4 < o ≤ 5, 5 - o, 0], B) → D'
  47. Segment(D, D') → h_1
  48. Polygon(A, B, D, C) → poly1, a_1, b_1, d_2, c_2
  49. Dilate(A, If[5.5 < o ≤ 6.5, 6.5 - o, 0], C) → A'_2
  50. Vector(A, A'_2) → w
  51. Translate(A, w) → A'_3
  52. Translate(A', w) → A''
  53. Dilate(A, If[6.5 < o ≤ 7.5, 7.5 - o, 0], B) → A'_4
  54. Vector(A, A'_4) → u_1
  55. Translate(A, u_1) → A'_5
  56. Translate(A'_1, u_1) → L
  57. Intersect(d, i) → M, N
  58. Intersect(d, g) → O, P
  59. Angle(A, C, N) → β
  60. Rotate(N, If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * β), (-β)], C) → N'
  61. Dilate(C, If[7.5 < o ≤ 8.5, 8.5 - o, 0], B) → C'_1
  62. Vector(C, C'_1) → v_1
  63. Ray(A, B) → q_1
  64. LineBisector(B, A) → r_1
  65. Distance(C, B) → distanceCB
  66. Segment(C, D) → i_1
  67. Line(A, D) → j_1
  68. Line(C, A) → m_1
  69. Intersect(e, j_1) → S, T
  70. Intersect(e, m_1) → U, V
  71. Angle(D, A, C) → γ
  72. Rotate(T, If[10 < o ≤ 11, ((o - 10) * γ), γ], A) → T'
  73. Midpoint(A, D) → A_1
  74. Midpoint(C, D) → Q
  75. Midpoint(A, B) → R
  76. Point(k_2) → Z
  77. Ray(Z, Q) → l_1
  78. Ray(B, C) → n_1
  79. Intersect(n_1, l_1) → B_1
  80. Ray(R, B_1) → p_1
  81. Intersect(t_1, p_1) → C_1
  82. Line(C_1, Q) → f_2
  83. Line(R, Z) → g_2
  84. Line(C, A) → h_2
  85. Intersect(h_2, t_1) → D_1
  86. Intersect(n_1, l_1) → E_1
  87. Segment(R, C_1) → i_2
  88. Dilate(A, If[6 < o ≤ 7, 7 - o, 0], R) → A'_6
  89. Segment(A, A'_6) → l_2
  90. Rotate(A, If[7 < o ≤ 8, ((-(o - 7)) * π), (-π)], R) → A'_7
  91. Rotate(A'_6, If[7 < o ≤ 8, ((-(o - 7)) * π), (-π)], R) → G_1
  92. Segment(A'_7, G_1) → m_2
  93. Dilate(A, If[8 < o ≤ 9, 9 - o, 0], C) → A'_8
  94. Vector(A, A'_8) → w_1
  95. Translate(A, w_1) → A'_9
  96. Translate(A'_6, w_1) → H_1
  97. Segment(A'_9, H_1) → n_2
  98. Translate(A'_7, w_1) → L_1
  99. Translate(G_1, w_1) → M_1
  100. Segment(L_1, M_1) → p_2
  101. Dilate(B, If[9 < o ≤ 10, 10 - o, 0], W) → B'_1
  102. Segment(B, B'_1) → j_2
  103. Dilate(W, If[10 < o ≤ 11, 11 - o, 0], C) → W'
  104. Segment(W, W') → q_2
  105. Rotate(B, If[11 < o ≤ 12, ((-(o - 11)) * π), (-π)], W) → B'_2
  106. Rotate(B'_1, If[11 < o ≤ 12, ((-(o - 11)) * π), (-π)], W) → I_1
  107. Segment(B'_2, I_1) → r_2
  108. Dilate(Z, If[13 < o ≤ 14, 14 - o, 0], Q) → Z'
  109. Segment(Z, Z') → s_2
  110. Dilate(Q, If[14 < o ≤ 15, 15 - o, 0], B_1) → Q'
  111. Segment(Q, Q') → t_2
  112. Dilate(C, If[14 < o ≤ 15, 15 - o, 0], B_1) → C'_2
  113. Segment(C, C'_2) → a_2
  114. Dilate(W, If[15 < o ≤ 16, 16 - o, 0], D_1) → W'_1
  115. Segment(W, W'_1) → b_2
  116. Line(W, D_1) → e_2
  117. Circle(D_1, 2) → c_3
  118. Intersect(c_3, e_2) → J_1, K_1
  119. Ray(D_1, K_1) → f_3
  120. Rotate(d_1, If[16 < o ≤ 17, ((o - 16) * π), π], C) → d_3
  121. Dilate(C, If[17 < o ≤ 18, 18 - o, 0], D_1) → C'_3
  122. Vector(C, C'_3) → u_2
  123. Translate(d_3, u_2) → e_3
  124. Intersect(g_2, j_2) → N_1
  125. Segment(R, N_1) → g_3
  126. Min(i_2, g_3) → o_1
  127. Circle(R, (0.35 * o_1)) → h_3
  128. Intersect(h_3, g_2) → O_1, P_1
  129. Angle(N_1, R, C_1) → δ
  130. Dilate(B_1, If[18 < o ≤ 19, 19 - o, 0], C_1) → Q_1
  131. Segment(B_1, Q_1) → i_3
  132. Dilate(C_1, If[19 < o ≤ 20, 20 - o, 0], R) → R_1
  133. Segment(C_1, R_1) → j_3
  134. Dilate(R, If[20 < o ≤ 21, 21 - o, 0], Z) → R'
  135. Segment(R, R') → k_3
  136. Dilate(C_1, If[21 < o ≤ 22, 22 - o, 0], Q) → S_1
  137. Segment(C_1, S_1) → l_3
  138. Rotate(P_1, If[22 < o ≤ 23, ((o - 22) * δ), δ], R) → T_1
  139. CircleSector(R, P_1, T_1) → p_3
  140. Dilate(R, If[23 < o ≤ 24, 24 - o, 0], C_1) → R'_1
  141. Vector(R, R'_1) → v_2
  142. Translate(p_3, v_2) → q_3

AI Drawing Prompt Reference

Draw a parallelogram ABCD with points A, B, C, and D as vertices. First, draw the line segment AB and the ray BC, then find the midpoint W of BC and draw a line t1 through W parallel to AB. Next, draw a line i from C to f, intersecting h at D. Connect BD and find point F1 on BD such that DF1 is parallel to t1. Then, draw angle BAC with degree α, and draw circles with centers at C, D, and radii 0.2 times BD, intersecting with corresponding lines to get points E, F, G, and H. Draw a circle with center at A and radius 0.15 times BD, intersecting with corresponding lines to get E', and draw a sector with center at A and radius E'E. Draw a circle with center at C and radius 0.2 times BD, intersecting with corresponding lines to get C', and draw a circle with center at C and radius C'C. Finally, draw a circle with center at D and radius 0.2 times BD, intersecting with corresponding lines to get D', and draw a circle with center at D and radius D'D.
The prompt above was compiled from the original drawing commands and is for reference only.