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Constructing a Parallelogram Using Symmetries

Constructing a Parallelogram Using Symmetries
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2018-11-01 19:38
PolygonTrigonometryComplex NumberGeometric Construction
This example demonstrates how to construct a parallelogram using symmetries. It involves core mathematical concepts such as the properties of parallelograms, symmetry, rotation, complex numbers, and trigonometry. The construction uses GeoGebra commands/objects like Segment, Line, Circle, Intersect, Polygon, Point, Rotate, Angle, and UnitVector. It is suitable for high school mathematics teaching or exploration by math enthusiasts.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Line(D, a) → b
  3. Circle(D, a) → c
  4. Intersect(c, b) → D_1, C
  5. Polygon(A, B, C, D) → poly1, a_1, b_1, c_1, d
  6. Segment(A, C) → e
  7. Segment(B, D) → f
  8. Intersect(f, e) → E
  9. Polygon(E, A, B, C, D, A, E) → poly2, e_1, a_2, b_2, c_2, d_1, a_3, e_2
  10. Point(Circle[F, e_1]) → G
  11. Rotate(G + (a_2, 0), Angle[B - A] + Angle[G - F] - Angle[A - E], G) → H
  12. Rotate(H + (b_2, 0), Angle[C - B] + Angle[H - G] - Angle[B - A], H) → I
  13. Rotate(I + (c_2, 0), Angle[D - C] + Angle[I - H] - Angle[C - B], I) → J
  14. Rotate(J + (d_1, 0), Angle[A - D] + Angle[J - I] - Angle[D - C], J) → K
  15. Rotate(K + (a_3, 0), Angle[E - A] + Angle[K - J] - Angle[A - D], K) → L
  16. Polygon(F, G, H, I, J, K, L) → poly3, f_1, g, h, i, j, k, l
  17. Intersect(i, h) → M
  18. Angle(K, J, I) → ε
  19. Angle(J, I, H) → ζ
  20. Angle(I, H, G) → η
  21. UnitVector(g) → u
  22. UnitVector(j) → v
  23. Angle(u, Vector[(-v)]) → θ
  24. Angle(D, C, B) → β
  25. Angle(C, B, A) → γ
  26. Angle(B, A, D) → δ
  27. Angle(A, D, C) → α
  28. Intersect(i, h) → N
  29. Segment(G, N) → m
  30. Intersect(j, i) → O
  31. Intersect(h, g) → Q
  32. Segment(O, Q) → n