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Symmedian Point and Tangential Triangle

Symmedian Point and Tangential Triangle
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2015-02-14 09:22
TriangleCircleGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates the symmedian point K as the center of perspective between the original triangle and its tangential triangle. Key math concepts include symmedian point, tangential triangle, perspective, and geometric construction. It is constructed using commands like Line, Midpoint, Segment, Intersect, Mirror, Circle, Center, Ray, Angle, Tangent, Polygon. Suitable for high school and above students to explore special points and triangle geometry.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. Midpoint(B, A) → D
  5. Midpoint(A, C) → E
  6. Midpoint(C, B) → F
  7. Segment(A, F) → d
  8. Segment(D, C) → e
  9. Segment(E, B) → f
  10. Intersect(d, e) → G
  11. Mirror(G, c) → G'_3
  12. Mirror(G, a) → G'_1
  13. Mirror(G, b) → G'_2
  14. Circle(G'_1, G'_3, G'_2) → g
  15. Center(g) → K
  16. Ray(A, K) → h
  17. Angle(G, A, E) → α
  18. Angle(B, A, K) → β
  19. Circle(A, C, B) → k
  20. Tangent(A, k) → i,
  21. Tangent(B, k) → j,
  22. Tangent(C, k) → l,
  23. Intersect(i, j) → C'
  24. Intersect(i, l) → B'
  25. Intersect(j, l) → A'
  26. Polygon(C', B', A') → poly1, j_1, h_1, i_1
  27. Line(A, A') → m
  28. Line(B, B') → n
  29. Line(C', C) → p

AI Drawing Prompt Reference

Draw a triangle ABC, then mark the midpoints D, E, F, and draw line segments d, e, f connecting AF, BD, CE. Find the intersection G of d and e, and reflect G to get G'1, G'2, G'3. Draw a circle g with G'1, G'3, G'2 as centers, find the center K, draw a ray h from A, find the angles GAE and BAK as α and β, draw the incircle k of triangle ABC, draw the tangents i, j, l to k, find the intersections C', B', A' of tangents i and j, i and l, j and l, and draw triangle C'B'A'. Finally, draw lines AA', BB', CC'.
The prompt above was compiled from the original drawing commands and is for reference only.