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Pedal Circles of Points on Feuerbach Hyperbola

Pedal Circles of Points on Feuerbach Hyperbola
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2015-02-16 09:05
Circle
In collections A Tour of Triangle Geometry
This example demonstrates that the pedal circle of any point on the Feuerbach hyperbola passes through the center of the hyperbola. It uses commands such as Line, Circle, Center, Centroid, Mirror, LineBisector, Polygon, OrthogonalLine, Intersect, AngularBisector, Point, Locus, Midpoint. Suitable for high school geometry learners and math enthusiasts.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. Circle(A, B, C) → d
  5. Center(d) → O
  6. Centroid(Polygon[A, B, C]) → G
  7. Mirror(G, c) → G'
  8. Mirror(G, a) → G'_1
  9. Mirror(G, b) → G'_2
  10. LineBisector(G'_1, G') → e
  11. LineBisector(G', G'_2) → f
  12. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  13. OrthogonalLine(A, b) → j
  14. OrthogonalLine(B, c) → k
  15. OrthogonalLine(C, a) → l
  16. Intersect(l, j) → H
  17. AngularBisector(B, A, C) → g
  18. AngularBisector(C, B, A) → h
  19. Intersect(g, h) → I
  20. Line(I, O) → i
  21. Point(i) → P
  22. Mirror(P, c) → P'_3
  23. Mirror(P, a) → P'_1
  24. Mirror(P, b) → P'_2
  25. LineBisector(P'_1, P'_3) → m
  26. LineBisector(P'_3, P'_2) → n
  27. Intersect(m, n) → P'
  28. Locus(P', P) → loc1
  29. Midpoint(B, C) → D
  30. Midpoint(C, A) → E
  31. Midpoint(A, B) → F_1
  32. Circle(F_1, E, D) → p
  33. OrthogonalLine(I, b) → q
  34. Intersect(b, q) → J
  35. Circle(I, J) → r
  36. Intersect(p, r, 1) → F
  37. OrthogonalLine(P', c) → s
  38. OrthogonalLine(P', b) → t
  39. OrthogonalLine(P', a) → d_1
  40. Intersect(d_1, a) → K
  41. Intersect(s, c) → L
  42. Intersect(t, b) → M
  43. Circle(K, L, M) → e_1