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Feuerbach Theorem and Feuerbach Point

Feuerbach Theorem and Feuerbach Point
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2015-02-14 06:03
TriangleCircleGeometric Construction
In collections A Tour of Triangle Geometry
This example illustrates the Feuerbach Theorem, which states that the nine-point circle is internally tangent to the incircle and externally tangent to the three excircles. Using GeoGebra commands such as Line, OrthogonalLine, LineBisector, etc., the nine-point circle, incircle, and excircles can be constructed and their relationships observed. Suitable for students at the junior high school level and above to learn about geometric constructions and properties of circles.

GeoGebra Commands & Parameters

  1. Line(A, B) → m
  2. Line(C, A) → n
  3. Line(B, C) → q
  4. OrthogonalLine(A, q) → a
  5. OrthogonalLine(B, n) → b
  6. OrthogonalLine(C, m) → c
  7. LineBisector(B, C) → d
  8. LineBisector(C, A) → e
  9. LineBisector(A, B) → f
  10. Intersect(q, a) → D
  11. Intersect(n, e) → E
  12. Intersect(n, b) → F_1
  13. Intersect(m, c) → G_1
  14. Intersect(m, f) → I
  15. Intersect(q, d) → J
  16. Circle(G_1, D, E) → k
  17. Segment(A, J) → i
  18. Segment(B, E) → j
  19. Segment(I, C) → l
  20. Intersect(i, j) → G
  21. AngularBisector(n, m) → g, h
  22. AngularBisector(q, m) → p, r
  23. AngularBisector(n, q) → s, t
  24. Intersect(g, p) → H
  25. Intersect(h, r) → K
  26. Intersect(h, p) → L
  27. Intersect(g, r) → M
  28. OrthogonalLine(M, q) → a_1
  29. OrthogonalLine(L, q) → b_1
  30. OrthogonalLine(H, n) → c_1
  31. Intersect(n, c_1) → N
  32. Circle(H, N) → d_1
  33. Intersect(q, a_1) → O
  34. Circle(M, O) → e_1
  35. Intersect(q, b_1) → P
  36. Circle(L, P) → f_1
  37. OrthogonalLine(K, q) → g_1
  38. Intersect(q, g_1) → Q
  39. Circle(K, Q) → h_1
  40. Intersect(h_1, k) → F, , ,