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GeoGebra Geometry Investigation: Halving

GeoGebra Geometry Investigation: Halving
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2023-03-22 03:12
TriangleCirclePolygonGeometric Construction
In collections Realidades de GeoGebra
This example demonstrates geometric investigations on halving shapes using GeoGebra. Core math concepts include area, constructions, division, squares, differential calculus, triangles, complex numbers, etc. Construction highlights the use of commands and objects like Intersect, Point, Line, OrthogonalLine, Circle, Polygon, Vector, Translate, Segment, Midpoint, Ray. Suitable for high school mathematics teaching or geometric construction learning.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Line(A, B) → a
  4. OrthogonalLine(A, a) → b
  5. Circle(A, B) → c
  6. Intersect(c, b, 2) → C
  7. OrthogonalLine(C, b) → d
  8. OrthogonalLine(B, a) → e
  9. Intersect(d, e) → D
  10. Polygon(A, B, D, C) → poly1, a_1, b_1, d_1, c_1
  11. Line(E, a_1) → f
  12. Point(f) → F
  13. Vector(E, F) → u
  14. OrthogonalLine(E, f) → g
  15. Circle(E, F) → h
  16. Intersect(h, g, 2) → G
  17. Vector(E, G) → v
  18. Translate(A, u) → A'
  19. Translate(B, u) → B'
  20. Translate(D, u) → D'
  21. Translate(C, u) → C'
  22. Polygon(A', B', D', C') → poly1', a', b', d', c'
  23. Translate(A', u) → A''
  24. Translate(B', u) → B''
  25. Translate(D', u) → D''
  26. Translate(C', u) → C''
  27. Polygon(A'', B'', D'', C'') → poly1'', a'', b'', d'', c''
  28. Translate(A, v) → A'_1
  29. Translate(B, v) → B'_1
  30. Translate(D, v) → D'_1
  31. Translate(C, v) → C'_1
  32. Polygon(A'_1, B'_1, D'_1, C'_1) → poly1'_1, a'_1, b'_1, d'_1, c'_1
  33. Translate(A', v) → A''_1
  34. Translate(B', v) → B''_1
  35. Translate(D', v) → D''_1
  36. Translate(C', v) → C''_1
  37. Polygon(A''_1, B''_1, D''_1, C''_1) → poly1''_1, a''_1, b''_1, d''_1, c''_1
  38. Translate(A'', v) → A'''
  39. Translate(B'', v) → B'''
  40. Translate(D'', v) → D'''
  41. Translate(C'', v) → C'''
  42. Polygon(A''', B''', D''', C''') → poly1''', a''', b''', d''', c'''
  43. Translate(D'_1, v) → D'_1'
  44. Translate(C'_1, v) → C'_1'
  45. Segment(D'_1', C'_1') → i
  46. Translate(D'_1, v) → D'_2
  47. Translate(C'_1, v) → C'_2
  48. Segment(D'_2, C'_2) → j
  49. Translate(A'_1, v) → A'_1'
  50. Translate(B'_1, v) → B'_1'
  51. Translate(D'_1, v) → D'_3
  52. Translate(C'_1, v) → C'_3
  53. Polygon(A'_1', B'_1', D'_3, C'_3) → poly1'_1', a'_1', b'_1', d'_3, c'_3
  54. Translate(A''_1, v) → A''_1'
  55. Translate(B''_1, v) → B''_1'
  56. Translate(D''_1, v) → D''_1'
  57. Translate(C''_1, v) → C''_1'
  58. Polygon(A''_1', B''_1', D''_1', C''_1') → poly1''_1', a''_1', b''_1', d''_1', c''_1'
  59. Translate(A''', v) → A''''
  60. Translate(B''', v) → B''''
  61. Translate(D''', v) → D''''
  62. Translate(C''', v) → C''''
  63. Polygon(A'''', B'''', D'''', C'''') → poly1'''', a'''', b'''', d'''', c''''
  64. Translate(A'', u) → A'''_1
  65. Translate(B'', u) → B'''_1
  66. Translate(D'', u) → D'''_1
  67. Translate(C'', u) → C'''_1
  68. Polygon(A'''_1, B'''_1, D'''_1, C'''_1) → poly1'''_1, a'''_1, b'''_1, d'''_1, c'''_1
  69. Translate(A''', u) → A''''_1
  70. Translate(B''', u) → B''''_1
  71. Translate(D''', u) → D''''_1
  72. Translate(C''', u) → C''''_1
  73. Polygon(A''''_1, B''''_1, D''''_1, C''''_1) → poly1''''_1, a''''_1, b''''_1, d''''_1, c''''_1
  74. Translate(A'''', u) → A'''''
  75. Translate(B'''', u) → B'''''
  76. Translate(D'''', u) → D'''''
  77. Translate(C'''', u) → C'''''
  78. Polygon(A''''', B''''', D''''', C''''') → poly1''''', a''''', b''''', d''''', c'''''
  79. Midpoint(i) → H
  80. Midpoint(a'_1') → I
  81. Polygon(H, D'_1', B'_1', I) → poly2, h_1, d'_1', b'_2, i_1
  82. Midpoint(d''_1') → J
  83. Segment(C''_1', J) → k
  84. Point(k) → K
  85. Circle(K, m) → p
  86. Intersect(p, d''_1', 1) → L
  87. OrthogonalLine(K, k) → l
  88. OrthogonalLine(L, d''_1') → n
  89. Intersect(a''_1', l) → M
  90. Intersect(a''_1', n) → N
  91. Polygon(M, N, L, K) → poly3, m_1, n_1, l_1, k_1
  92. Midpoint(d'_1) → O
  93. Polygon(A'_1, O, B'_1) → poly4, b'_3, a'_2, o
  94. Point(d''_1) → P
  95. Polygon(A''_1, P, B''_1) → poly5, b''_2, a''_2, p_1
  96. Midpoint(d_1) → Q
  97. Midpoint(a_1) → R
  98. Polygon(A, Q, D, R) → poly6, a_2, q, d_2, r
  99. Midpoint(d') → S
  100. Midpoint(a') → T
  101. Midpoint(a_1) → U
  102. Segment(C', S) → s
  103. Segment(T, A') → t
  104. Point(s) → V
  105. Point(t) → W
  106. Circle(V, m) → e_1
  107. Circle(W, m) → f_1
  108. Polygon(W, V, Z, A_1) → poly7, w, v_1, z, a_3
  109. Midpoint(D'', A'') → E_1
  110. Midpoint(A'', E_1) → F_1
  111. Midpoint(E_1, D'') → G_1
  112. Segment(C'', B'') → j_1
  113. Polygon(C'', F_1, B'', G_1) → poly8, c''_2, f_2, b''_3, g_2
  114. Midpoint(d'''_1) → H_1
  115. Midpoint(b'''_1) → I_1
  116. Midpoint(a'''_1) → J_1
  117. Midpoint(c'''_1) → K_1
  118. Segment(K_1, J_1) → q_1
  119. Segment(I_1, H_1) → r_1
  120. Point(q_1) → L_1
  121. Point(r_1) → M_1
  122. Polygon(C'''_1, L_1, B'''_1, M_1) → poly9, c'''_2, l_2, b'''_2, m_2
  123. Midpoint(d'''') → B_1
  124. Midpoint(c'''') → C_1
  125. Midpoint(a'''') → D_1
  126. Midpoint(b'''') → N_1
  127. Polygon(B_1, N_1, D_1, C_1) → poly10, b_2, n_2, d_3, c_2
  128. Point(d''''') → O_1
  129. Point(c''''') → P_1
  130. OrthogonalLine(P_1, c''''') → g_1
  131. OrthogonalLine(O_1, d''''') → s_1
  132. Intersect(a''''', s_1) → Q_1
  133. Intersect(b''''', g_1) → R_1
  134. Polygon(P_1, Q_1, R_1, O_1) → poly11, p_2, q_2, r_2, o_1
  135. Midpoint(d''') → S_1
  136. Midpoint(a''') → T_1
  137. Midpoint(S_1, D''') → U_1
  138. Midpoint(T_1, A''') → V_1
  139. Polygon(U_1, V_1, B''', D''') → poly12, u_1, v_2, b'''_3, d'''_2
  140. Point(d''''_1) → W_1
  141. Midpoint(A''''_1, D''''_1) → Z_1
  142. Ray(W_1, Z_1) → t_1
  143. Intersect(a''''_1, t_1) → A_2
  144. Polygon(A_2, B''''_1, D''''_1, W_1) → poly13, a_4, b''''_2, d''''_2, w_1

AI Drawing Prompt Reference

On a coordinate plane, start with a right triangle whose vertices are the origin, a point on the x-axis, and a third point found by drawing a circle centered at the origin through the x-axis point, then intersecting it with a line perpendicular to the hypotenuse through the origin. Complete this into a square. Next, construct a ray outside the square, pick a point on it, and use the vector from the origin to that point to translate the square twice, creating a row of three squares. Then, use a perpendicular vector to translate this entire row twice, forming a 3-by-3 grid of squares. Finally, connect two translated vertices at the grid's edge with two parallel segments, illustrating a halving construction.
The prompt above was compiled from the original drawing commands and is for reference only.