# zbMATH — the first resource for mathematics

A projective generalization of the Droz-Farny line theorem. (English) Zbl 1076.51008
The Droz-Farny line theorem states that if two perpendicular lines $$\alpha$$ and $$\beta$$ are drawn through the orthocenter of a triangle $$ABC$$, then the midpoints $$A'$$, $$B'$$, and $$C'$$ of the three line segments that $$\alpha$$ and $$\beta$$ intercept on the sidelines of $$ABC$$ are collinear. In this paper, the authors give and prove a projective generalization of this theorem.
A proof of the Droz-Farny line theorem, together with a historical commentary, is given by J-L. Ayme in [Forum Geom. 4, 219–224 (2004; Zbl 1078.51019)]. Neuberg’s description of the envelope of the line $$A'B'C'$$ as $$\alpha$$ and $$\beta$$ rotate is given by R. Goormaghtigh in [Solution II to Problem 3758, Am. Math. Mon. 44, 668–672 (1937)], and an investigation of the situation when $$\alpha$$ and $$\beta$$ are any transversals (not necessarily perpendicular) through any point (not necessarily the orthocenter) is made by S. T. Kao and A. Bernhart in [Am. Math. Mon. 72, 718–725 (1965; Zbl 0129.12802)].

##### MSC:
 51M04 Elementary problems in Euclidean geometries
Full Text: