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Wallace-Simson Lines of Antipodal Points

Wallace-Simson Lines of Antipodal Points
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2015-02-15 02:42
TriangleCircleGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates that the Simson lines of a point and its antipodal point on the circumcircle intersect perpendicularly on the nine-point circle. It covers the concepts of circle, antipodal points, and Simson lines. The construction uses commands like Line, Circle, Point, OrthogonalLine, Intersect, Center, Mirror, Midpoint, and Angle. Suitable for high school geometry education.

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. Point(d) → P
  6. OrthogonalLine(P, a) → e
  7. OrthogonalLine(P, c) → f
  8. OrthogonalLine(P, b) → g
  9. Intersect(a, e) → P_c
  10. Intersect(c, f) → P_b
  11. Intersect(b, g) → P_a
  12. Line(P_c, P_b) → h
  13. OrthogonalLine(A, b) → j
  14. OrthogonalLine(B, c) → k
  15. Intersect(j, k) → H
  16. Center(d) → D
  17. Mirror(P, D) → P'
  18. OrthogonalLine(P', b) → i
  19. OrthogonalLine(P', a) → l
  20. OrthogonalLine(P', c) → m
  21. Intersect(l, a) → E
  22. Intersect(b, i) → F
  23. Intersect(c, m) → G
  24. Line(E, G) → n
  25. Midpoint(B, C) → I
  26. Midpoint(C, A) → J
  27. Midpoint(A, B) → K
  28. Circle(K, J, I) → p
  29. Angle(h, n) → α

AI Drawing Prompt Reference

Draw a triangle ABC, and construct a circle with A, B, and C as its center points. Choose a point P on the circle and draw the pedal triangle P_aP_bP_c with respect to the sides of the triangle. Then, find the symmetric point P' of P with respect to the circle's center and draw the pedal triangle P''_aP''_bP''_c with respect to the sides of the triangle. Next, find the intersection of P_cP_b and P''_aP''_b, and the intersection of P_aP_b and P''_aP''_c, and the intersection of P_bP_c and P''_bP''_c, which form the nine-point circle.
The prompt above was compiled from the original drawing commands and is for reference only.