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Symmedian Point: An Important Geometric Concept

Symmedian Point: An Important Geometric Concept
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2015-02-12 06:41
TriangleGeometric ConstructionLocus
In collections A Tour of Triangle Geometry
This example demonstrates the Symmedian Point in a triangle, which is isogonal to the centroid. It involves geometric construction using tools like Segment, AngularBisector, and OrthogonalLine in GeoGebra. The concept is useful for understanding triangle properties and symmetry. Suitable for high school students or college students studying geometry.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Segment(B, C) → b
  3. Segment(C, A) → c
  4. AngularBisector(A, B, C) → d
  5. AngularBisector(C, A, B) → e
  6. AngularBisector(B, C, A) → f
  7. OrthogonalLine(B, d) → l
  8. OrthogonalLine(A, e) → m
  9. OrthogonalLine(C, f) → a_1
  10. Polygon(A, C, 4) → poly1, g, h, i, j, D, E
  11. Polygon(C, B, 4) → poly2, k, n, p, q, F, J
  12. Polygon(B, A, 4) → poly3, r, s, t, b_1, H, I
  13. Line(I, H) → c_1
  14. Line(E, D) → d_1
  15. Line(J, F) → e_1
  16. Midpoint(b) → J_1
  17. Midpoint(c) → P
  18. Midpoint(a) → L
  19. Segment(A, J_1) → f_1
  20. Segment(L, C) → g_1
  21. Segment(P, B) → h_1
  22. Intersect(f_1, g_1) → G
  23. Intersect(c_1, d_1) → M
  24. Intersect(c_1, e_1) → N
  25. Intersect(d_1, e_1) → O
  26. Line(M, A) → i_1
  27. Line(N, B) → j_1
  28. Line(O, C) → k_1
  29. Intersect(i_1, j_1) → K

AI Drawing Prompt Reference

Draw a triangle and construct the angle bisectors of each angle. From each vertex, draw perpendicular lines to the opposite sides, meeting at the orthocenter. Then, create squares on each side of the triangle using the vertices and the orthocenter. Finally, connect the midpoints of the opposite sides of these squares to explore their relationship with the orthocenter.
The prompt above was compiled from the original drawing commands and is for reference only.