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.

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