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Orthic Triangle in a 120-Degree Angle Triangle

Orthic Triangle in a 120-Degree Angle Triangle
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2017-02-13 22:53
TriangleCircleGeometric Construction
This example demonstrates the construction of an orthic triangle within a triangle containing a 120-degree angle. Key mathematical concepts include the orthocenter, trigonometric functions, plane figures, circles, and vectors. The construction utilizes GeoGebra commands such as Rotate, CircumcircleArc, Point, Segment, OrthogonalLine, Intersect, Ray, Polygon, Circle, Angle, and Midpoint. It is suitable for high school mathematics education or for geometry enthusiasts exploring triangle properties.

GeoGebra Commands & Parameters

  1. Rotate(B, 120°, A_1) → C
  2. CircumcircleArc(B, A_1, C) → c
  3. Point(c) → A
  4. Segment(B, C) → f
  5. OrthogonalLine(A, f) → g
  6. Intersect(g, f) → D
  7. Ray(B, A) → h
  8. Ray(C, A) → i
  9. OrthogonalLine(C, h) → j
  10. OrthogonalLine(B, i) → k
  11. Intersect(j, h) → E
  12. Intersect(k, i) → F
  13. Polygon(F, D, E) → poly1, e, f_1, d
  14. Intersect(j, k) → H
  15. OrthogonalLine(F, f_1) → l
  16. OrthogonalLine(D, d) → m
  17. Intersect(l, m) → H'
  18. Segment(H', A) → n
  19. Segment(H', H) → p
  20. Polygon(A, C, B) → poly2, b, a, c_1
  21. Circle(A, H', H) → q
  22. Segment(A, E) → r
  23. Segment(H, A) → t
  24. Segment(F, A) → s
  25. Segment(F, H') → g_1
  26. Segment(H', E) → h_1
  27. Angle(B, E, C) → α
  28. Angle(B, F, C) → β
  29. Midpoint(A, H) → S