Steiner's Ellipse

Let $\alpha':\beta':\gamma'$ be the Isotomic Conjugate Point of a point with Trilinear Coordinates $\alpha:\beta:\gamma$. The isotomic conjugate of the Line at Infinity having trilinear equation

$$a\alpha+b\beta+c\gamma=0$$

is

$${\beta'\gamma'\over a}+{\gamma'\alpha'\over b}+{\alpha'\beta'\over c}=0,$$

known as Steiner's ellipse (Vandeghen 1965).

See also Isotomic Conjugate Point, Line at Infinity

References

Vandeghen, A. ``Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Triangle.'' Amer. Math. Monthly 72, 1091-1094, 1965.