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Orthocenter and Circumcenter Relationship

Orthocenter and Circumcenter Relationship
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2017-10-04 05:06
TriangleCircleGeometric Construction
This example demonstrates the geometric relationship and positional relationship between the orthocenter and circumcenter of a triangle. It involves core mathematical concepts such as the orthocenter, circumcenter, medians, angle bisectors, circles, lines, and function graphs. The construction highlights the use of GeoGebra commands and objects like Circle, AngularBisector, Center, Mirror, Line, Intersect, Segment, and Angle. It is suitable for high school mathematics education to understand the properties and mutual relationships of important geometric points in a triangle.

GeoGebra Commands & Parameters

  1. Circle(A, B, C) → c
  2. AngularBisector(A, C, B) → f
  3. AngularBisector(C, B, A) → g
  4. AngularBisector(B, A, C) → h
  5. Center(c) → O
  6. Mirror(O, f) → P'_3
  7. Mirror(O, g) → P'_1
  8. Mirror(O, h) → P'_2
  9. Line(A, B) → p
  10. Line(B, C) → q
  11. Line(C, A) → r
  12. Line(C, P'_3) → l
  13. Line(B, P'_1) → m
  14. Line(A, P'_2) → n
  15. Intersect(l, m) → H
  16. Segment(C, O) → i
  17. Segment(A, O) → k
  18. Segment(B, O) → j
  19. Intersect(l, p) → C'
  20. Intersect(n, q) → A'
  21. Intersect(m, r) → B'
  22. Segment(A, A') → s
  23. Segment(B, B') → t
  24. Segment(C, C') → a
  25. Angle(C, C', A) → α
  26. Angle(A, A', B) → β
  27. Angle(B, B', C) → γ
  28. Segment(A, B) → b
  29. Segment(B, C) → d
  30. Segment(C, A) → e
  31. Angle(A, C', C) → δ
  32. Angle(C, B', B) → ε
  33. Angle(B, A', A) → ζ