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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
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