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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
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