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Metric Relationships Proven Briefly

Metric Relationships Proven Briefly
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2020-01-23 21:27
PolygonVectorTrigonometryCoordinate GeometryGeometric Construction
This example dynamically demonstrates metric relationships in geometric figures like rectangles and triangles, revealing intrinsic mathematical principles. Core concepts include coordinate geometry, trigonometric functions, vector operations, and geometric transformations. Construction highlights the use of GeoGebra commands/objects such as Line, Point, Circle, Segment, Intersect, OrthogonalLine, Polygon, Vector, Translate, Midpoint, and Angle. Suitable for high school and above, it bridges geometric proofs with dynamic geometry.

GeoGebra Commands & Parameters

  1. Line(O, xAxis) → b
  2. Point(b) → E
  3. Circle(O, a) → c
  4. Point(c) → A
  5. Segment(A, E) → d2
  6. Circle(A, If[t < d2, t, d2]) → e2
  7. Intersect(c, b) → C, B
  8. Segment(A, C) → e
  9. OrthogonalLine(E, b) → i
  10. Circle(E, e) → k
  11. Intersect(k, i) → G, I
  12. Segment(C, I) → b2
  13. Circle(C, If[t < b2, t, b2]) → c2
  14. Segment(A, E) → t1
  15. Line(A, E) → h1
  16. Segment(A, E) → f1
  17. Circle(A, If[t < f1, t, f1]) → g1
  18. Intersect(g1, h1) → U, V
  19. Line(C, I) → n
  20. Segment(C, I) → d1
  21. Circle(C, If[t < t1, t, t1]) → p
  22. Segment(A, B) → f
  23. Circle(E, f) → h
  24. Intersect(h, b) → D, F
  25. Segment(B, F) → i1
  26. Circle(B, If[t < i1, t, i1]) → k1
  27. OrthogonalLine(A, b) → d
  28. Intersect(d, b) → H
  29. Segment(A, H) → g
  30. Segment(I, F) → j
  31. Segment(I, E) → l
  32. Segment(E, F) → m
  33. Line(A, b) → q
  34. Intersect(q, i) → L
  35. Segment(H, B) → r
  36. Circle(E, r) → s
  37. Intersect(s, b) → M, N
  38. Segment(L, N) → a1
  39. Circle(E, g) → c1
  40. Intersect(c1, b) → J, K
  41. Segment(C, H) → b1
  42. Circle(E, b1) → e1
  43. Intersect(e1, i) → P, Q
  44. Polygon(I, E, F) → poly1, f_1, i_1, e_1
  45. Polygon(A, C, B) → poly2, b_1, a_1, c_1
  46. Polygon(A, H, B) → poly3, b_2, a_2, h_1
  47. Polygon(L, E, N) → poly4, n_1, l_1, e_2
  48. Polygon(A, C, H) → poly5, h_2, a_3, c_2
  49. Polygon(Q, E, K) → poly6, k_1, q_1, e_3
  50. Intersect(f1, h1) → T
  51. Intersect(k1, t1) → Z, H1
  52. Segment(A, L) → j1_1
  53. Circle(A, If[t < j1_1, t, j1_1]) → p1
  54. Intersect(p1, j1_1) → A1, B1
  55. Vector(A, A1) → u
  56. Translate(A, u) → A'
  57. Translate(H, u) → H'
  58. Translate(B, u) → B'
  59. Polygon(A', H', B') → poly3', b', a', h'
  60. Segment(A, K) → m1_2
  61. Circle(A, If[t < m1_2, t, m1_2]) → q1
  62. Intersect(q1, m1_2) → C1, D1
  63. Segment(C, Q) → l1_2
  64. Circle(C, If[t < l1_2, t, l1_2]) → r1
  65. Intersect(r1, l1_2, 1) → E1
  66. Segment(H, E) → n1
  67. Circle(H, If[t < n1, t, n1]) → s1
  68. Intersect(s1, b) → F1, G1
  69. Polygon(C1, E1, G1) → poly8, g1_1, c1_1, e1_1
  70. Segment(B, F) → j1
  71. Intersect(k1, j1) → I1, J1
  72. Intersect(c2, b2) → S, R
  73. Intersect(e2, d2) → W, K1
  74. Segment(S, W) → l1
  75. Segment(W, I1) → m1
  76. Segment(I1, S) → a2
  77. Polygon(S, W, I1) → poly7, i1_1, s_1, w
  78. Circle(H, a / 10) → f2
  79. Intersect(f2, r) → L1,
  80. Intersect(f2, b1) → M1,
  81. Intersect(f2, g) → N1,
  82. Circle(A, a / 10) → g2
  83. Intersect(g2, f) → O1,
  84. Intersect(g2, e) → P1,
  85. Line(O1, e) → h2
  86. Line(P1, f) → i2
  87. Intersect(i2, h2) → Q1
  88. Segment(O1, Q1) → j2
  89. Segment(Q1, P1) → k2
  90. Line(M1, g) → l2
  91. Line(L1, g) → m2
  92. Line(N1, r) → n2
  93. Intersect(l2, n2) → R1
  94. Intersect(m2, n2) → S1
  95. Polygon(R1, M1, H, N1) → poly9, r1_1, m1_1, h_3, n1_1
  96. Polygon(N1, S1, L1, H) → poly10, n1_2, s1_1, l1_1, h_4
  97. Circle(E, a / 10) → p2
  98. Intersect(p2, c1_1) → U1,
  99. Line(U1, r) → r2
  100. Intersect(p2, e1_1) → V1,
  101. Line(V1, g) → q2
  102. Line(U1, r) → s2
  103. Intersect(s2, q2) → W1
  104. Polygon(U1, W1, V1, E) → poly11, u1, w1, v1, e_4
  105. Midpoint(w) → Z1
  106. Midpoint(h') → A2
  107. Midpoint(g1_1) → B2
  108. Polygon(C2, D2, E2, F2) → poly12, c2_1, d2_1, e2_1, f2_1
  109. Polygon(G2, H2, I2, J2) → poly13, g2_1, h2_1, i2_1, j2_1
  110. Vector(K2, L2) → v
  111. Vector(M2, N2) → z
  112. Vector(O2, P2) → z1
  113. Polygon(Q2, R2, S2, U2) → poly14, q2_1, r2_1, s2_1, u2
  114. Vector(E, A_1) → u_1
  115. Vector(E, B_1) → v_1
  116. Vector(H, C_1) → w_1
  117. Translate(W, Vector[(t / 10 * w_1)]) → W'
  118. Translate(A, Vector[(t / 10 * u_1)]) → A'_1
  119. Translate(H, Vector[(t / 10 * u_1)]) → H'_1
  120. Translate(B, Vector[(t / 10 * u_1)]) → B'_1
  121. Polygon(A'_1, H'_1, B'_1) → poly3'_1, b'_1, a'_1, h'_1
  122. Translate(A, Vector[(t / 10 * v_1)]) → A'_2
  123. Translate(C, Vector[(t / 10 * v_1)]) → C'
  124. Translate(H, Vector[(t / 10 * v_1)]) → H'_2
  125. Polygon(A'_2, C', H'_2) → poly5', h'_2, a'_2, c'
  126. Translate(A, Vector[(t / 10 * w_1)]) → A'_3
  127. Translate(C, Vector[(t / 10 * w_1)]) → C'_1
  128. Translate(B, Vector[(t / 10 * w_1)]) → B'_2
  129. Polygon(A'_3, C'_1, B'_2) → poly2', b'_2, a'_3, c'_1
  130. Segment(C'_1, B'_2) → d_1
  131. Angle(C'_1, A'_3, B'_2) → α
  132. Angle(A'_2, H'_2, C') → β
  133. Angle(B'_1, H'_1, A'_1) → γ
  134. Segment(C'_1, B'_2) → g_1
  135. Segment(A'_2, C') → j_1
  136. Segment(A'_1, B'_1) → m_1
  137. Segment(A'_1, H'_1) → p_1
  138. Segment(H'_1, B'_1) → r_1
  139. Segment(A'_3, B'_2) → t_1
  140. Segment(A'_3, C'_1) → d_2
  141. Segment(A'_2, H'_2) → f_2
  142. Segment(C', H'_2) → g_2
  143. Angle(A'_2, H'_2, C') → δ
  144. Angle(B'_1, H'_1, A'_1) → ε

AI Drawing Prompt Reference

Draw a dynamic geometric figure demonstrating metric relationships in rectangles and triangles. Include coordinate axes, circles, line segments, and points, constructed through intersections and perpendicular lines. Use vector operations and geometric transformations to reveal intrinsic mathematical principles, ultimately forming multiple polygons to show the connection between dynamic geometry and geometric proofs.
The prompt above was compiled from the original drawing commands and is for reference only.