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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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