# zbMATH — the first resource for mathematics

The Droz-Farny theorem and related topics. (English) Zbl 1119.51020
Summary: 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.

##### MSC:
 51M05 Euclidean geometries (general) and generalizations
Full Text: