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Steiner Chain

Steiner Chain
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2025-08-19 02:04
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A Steiner chain is a special type of geometric figure used to demonstrate the Steiner construction of polygons. This example shows how to construct a Steiner chain using GeoGebra, illustrating the basic principles of the Steiner construction of polygons. Key mathematical concepts include geometric construction, polygons, and Steiner construction. Construction points include using Circle, Intersect

GeoGebra Commands & Parameters

  1. Circle((0, 0), 1) → c
  2. Intersect(Circle[((1 - (sin(pi / p) / cos(pi / q))^(2))^(-1 / 2), 0), ((cos(pi / q) / sin(pi / p))^(2) - 1)^(-1 / 2)], Ray[(0, 0), Rotate[(1, 0), pi / p]], 1) → P0
  3. Sequence(Rotate[P0, ((k * 2) * pi / p)], k, 0, p - 1) → V
  4. Circle(Mirror[V(p), c], V(p), V(1)) → hc1
  5. Polygon(pV) → actualp
  6. Flatten(Join[{{Vac(p)}, {Vac \ {Vac(p)}}}]) → Vacp
  7. Flatten(Join[{{Vac \ {Vac(1)}}, {Vac(1)}}]) → Vac2
  8. Zip(Center[a], a, s) → S
  9. Circle(((-0.22475, -0.22475) + (0.63683, 0.63683)) / 2, (-0.22475, -0.22475)) → cExt
  10. Circle(((0.22475, 0.22475) + (0.47618, 0.47618)) / 2, (0.22475, 0.22475)) → cInt
  11. Conic(S(1), S(2), S(3), S(4), S(5)) → e1
  12. Intersect(hc1, Ray[(V(1) + V(p)) / 2, (0, 0)]) → V0,
  13. CircumcircleArc(V(p), V0, V(1)) → arcoCelda
  14. Join(Sequence[Rotate[Sequence[Point[arcoCelda, s], s, 0, 0.9, 0.1], ((2 * k) * pi / p)], k, 0, p - 1]) → celda0
  15. Polygon(celda0) → pol
  16. Sequence({Vac(k), q - 1, ncActual}, k, 3, p - 1) → Vplus
  17. If(floor(p / 2) ≠ p / 2 ∧ q ≟ 3, 4, floor(q / 2) ≟ q / 2, 2, 3) → nc
  18. Sequence({V(k), q - 1, 0}, k, 1, p) → vert0
  19. Flatten(Join[{{Vac \ {Vac(1), Vac(2)}}, {Vac(1), Vac(2)}}]) → Vac2'
  20. Flatten(Join[{{Vac(p - 1), Vac(p)}, {Vac \ {Vac(p - 1), Vac(p)}}}]) → Vacp'