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Isotomic Conjugate of Point at Infinity and Steiner Ellipse

Isotomic Conjugate of Point at Infinity and Steiner Ellipse
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2015-02-14 22:03
CircleCoordinate GeometryGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates that the isotomic conjugate of the point at infinity lies on the Steiner circumellipse with the centroid as its center. Using GeoGebra commands such as Line, Circle, Point, Segment, AngularBisector, etc., the Steiner ellipse can be constructed and its properties observed. Suitable for high school geometry learners and math enthusiasts.

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. Segment(P, A) → e
  7. Segment(P, B) → f
  8. Segment(P, C) → g
  9. AngularBisector(B, A, C) → h
  10. AngularBisector(B, C, A) → i
  11. AngularBisector(A, B, C) → j
  12. Mirror(P, h) → P'
  13. Mirror(P, i) → P'_1
  14. Mirror(P, j) → P'_2
  15. Line(A, P') → k
  16. Line(P'_1, C) → l
  17. Line(P'_2, B) → m
  18. Intersect(b, k) → D
  19. Intersect(c, m) → E
  20. Intersect(a, l) → F
  21. Midpoint(A, B) → J
  22. Midpoint(A, C) → H
  23. Midpoint(C, B) → I
  24. Mirror(E, H) → E'
  25. Mirror(F, J) → F'
  26. Mirror(D, I) → D'
  27. Line(A, D') → n
  28. Line(B, E') → p
  29. Line(C, F') → q
  30. Intersect(n, p) → S
  31. Locus(S, P) → loc1
  32. Point(loc1) → K
  33. Point(loc1) → L
  34. Point(loc1) → M
  35. Point(loc1) → N
  36. Conic(K, L, M, N, C) → r
  37. Center(r) → G

AI Drawing Prompt Reference

Draw a triangle ABC and construct its circumcircle d. Choose a point P on circle d, and construct the feet of the perpendiculars from P to the sides of the triangle, P', P'_1, and P'_2. Draw lines AP', BP'_1, and CP'_2, which intersect BC, CA, and AB at D, E, and F, respectively. Find the midpoints J, H, and I of AB, AC, and BC, and construct the lines n, p, and q, which are the reflections of DE, EF, and FD over H, I, and J, intersecting BC, CA, and AB at S. Construct the locus of S as point P moves on circle d, and select four points K, L, M, and N from this locus. Use K, L, M, N, and C as vertices to construct an ellipse r, and find the center G of this ellipse.
The prompt above was compiled from the original drawing commands and is for reference only.