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Isogonal Conjugate of Centroid is Symmedian Point

Isogonal Conjugate of Centroid is Symmedian Point
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2015-02-14 09:13
TriangleCoordinate GeometryGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates that the isogonal conjugate of the centroid is the symmedian point. Key mathematical concepts include the centroid and symmedian point of a triangle. Construction highlights the use of line, midpoint, segment, intersect, mirror, circle, center, ray, angle, polygon commands. 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. 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. Polygon(A, C, 4) → poly1, i, j, k, l, H, I
  20. Polygon(C, B, 4) → poly2, m, n, p, q, J, L
  21. Polygon(B, A, 4) → poly3, r, s, t, a_1, M, N
  22. Line(N, M) → b_1
  23. Line(I, H) → c_1
  24. Line(L, J) → d_1
  25. Intersect(b_1, c_1) → O
  26. Intersect(b_1, d_1) → P
  27. Intersect(c_1, d_1) → Q
  28. Line(O, A) → e_1
  29. Line(P, B) → f_1
  30. Line(C, Q) → g_1

AI Drawing Prompt Reference

Draw a triangle and label its centroid and symmedian point, demonstrating that the isogonal conjugate of the centroid is the symmedian point. Start by drawing the triangle's sides and finding the midpoints of each side. Connect the vertices to their respective midpoints. Locate the intersection points of these segments, then reflect these points across each side to get three new points. Connect these three points to form a circle, whose center is the symmedian point. Draw a ray from a vertex to the center and label the relevant angles and auxiliary constructions to clearly illustrate the relationship between the centroid and the symmedian point.
The prompt above was compiled from the original drawing commands and is for reference only.