Charles Thas, The Droz-Farny Theorem and Related Topics,
Forum Geometricorum, 6 (2006) 235--240.
Abstract: At each point P of the Euclidean plane Pi , not on the sidelines
of a triangle A_1A_2A_3 of Pi , there exists an involution in the pencil
of lines through P, such that each pair of conjugate lines intersect the
sides of A_1A_2A_3 in segments with collinear midpoints. If P=H, the orthocenter
of A_1A_2A_3, this involution becomes the orthogonal involution (where orthogonal
lines correspond) and we find the well-known Droz-Farny Theorem, which says
that any two orthogonal lines through $H$ intersect the sides of the triangle
in segments with collinear midpoints. In this paper we investigate
two closely related loci that have a strong connection with the Droz-Farny
Theorem. Among examples of these loci we find the circumcirle of the anticomplementary
triangle and the Steiner ellipse of that triangle.