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A purely synthetic proof of the Droz-Farny line theorem. (English) Zbl 1078.51019
The author gives a short, elegant and convincing proof of the Droz-Farny line theorem which says: Let \(H\) be the orthocenter of a triangle \(\Delta\) of the real Euclidean plane and \((\ell,\ell^{\bot})\) be a pair of orthogonal lines through \(H\), then \((\ell,\ell^{\bot})\) intercept on the three triangle sides \(a,b,c\) three segments whose midpoints are collinear.
The proof is based on Miquel’s pivot theorem and on the following two facts concerning the reflections \(\rho_a\), \(\rho_b\), \(\rho_c\) about \(a,b,c\), respectively:
1. By L. N. M. Carnot (1801), the three points \(\rho_a(H)\), \(\rho_b(H)\), \(\rho_c(H)\) belong to the circumcircle \(C\) of \(\Delta\).
2. The three lines \(\rho_a(\ell)\), \(\rho_b(\ell)\), \(\rho_c(\ell)\) are concurrent and their common point is on \(C\).

MSC:
51M04 Elementary problems in Euclidean geometries
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