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Orthocenter Symmetry Demo

Orthocenter Symmetry Demo
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2017-10-22 04:27
TriangleCoordinate GeometryGeometric Construction
This example demonstrates the geometric properties of the orthocenter and symmetry in triangles. It involves core mathematical concepts like orthocenter, symmetry transformation, similarity transformation, and coordinate geometry. The construction utilizes GeoGebra commands and objects such as Circle, Point, Polygon, Angle, Line, OrthogonalLine, Intersect, Segment, and Mirror. It is suitable for high school and above students to understand the properties of the orthocenter and applications of symmetry transformations.

GeoGebra Commands & Parameters

  1. Circle((0, 0), 2.5) → c_2
  2. Point(c_2) → A
  3. Point(c_2) → B
  4. Point(c_2) → C
  5. Polygon(A, B, C) → t1, c, a, b
  6. Angle(C, A, B) → α
  7. Angle(A, B, C) → β
  8. Angle(B, C, A) → γ
  9. Line(B, C) → f_a
  10. Line(A, C) → f_b
  11. Line(A, B) → f_c
  12. OrthogonalLine(A, f_a) → fh_a
  13. OrthogonalLine(B, f_b) → fh_b
  14. OrthogonalLine(C, f_c) → fh_c
  15. Intersect(f_a, fh_a) → D
  16. Intersect(f_b, fh_b) → E
  17. Intersect(f_c, fh_c) → F
  18. Intersect(fh_a, fh_b) → H
  19. Segment(C, F) → h_c
  20. Segment(A, D) → h_a
  21. Segment(B, E) → h_b
  22. Segment(A, H) → h_{aH}
  23. Segment(B, H) → h_{bH}
  24. Segment(C, H) → h_{cH}
  25. Segment(A, F) → l_1
  26. Segment(A, E) → m_1
  27. Segment(C, D) → r_1
  28. Segment(C, E) → m_2
  29. Segment(D, B) → r_2
  30. Segment(B, F) → l_2
  31. Intersect(c_2, fh_a, 2) → H_a
  32. Intersect(c_2, fh_b, 2) → H_b
  33. Intersect(c_2, fh_c, 2) → H_c
  34. Segment(H_c, B) → bh_c
  35. Segment(B, H_a) → bh_a
  36. Segment(H_a, C) → ch_a
  37. Segment(C, H_b) → ch_b
  38. Segment(H_b, A) → ah_b
  39. Segment(A, H_c) → ah_c
  40. Segment(E, H_b) → bh_b
  41. Segment(D, H_a) → ah_a
  42. Segment(F, H_c) → ch_c
  43. Circle(B, H) → c_b
  44. Circle(C, H) → c_c
  45. Circle(A, H) → c_a
  46. Mirror(c_2, a) → d
  47. Mirror(c_2, b) → e
  48. Mirror(c_2, c) → f

AI Drawing Prompt Reference

Draw a triangle ABC, then construct its orthocenter H. Use perpendicular lines from each vertex to the opposite sides to construct the altitudes h_a, h_b, and h_c. Next, find the symmetric points H_a, H_b, and H_c of H with respect to the sides of the triangle. Finally, draw three circles with these symmetric points as centers and the side lengths of the triangle as radii.
The prompt above was compiled from the original drawing commands and is for reference only.