Droz-Farny Line Theorem
In Euclidean geometry, the Droz-Farny line theorem is a property of two perpendicular lines through the orthocenter of an arbitrary triangle. Let T be a triangle with vertices A, B, and C, and let H be its orthocenter (the common point of its three altitude lines. Let L_1 and L_2 be any two mutually perpendicular lines through H. Let A_1, B_1, and C_1 be the points where L_1 intersects the side lines BC, CA, and AB, respectively. Similarly, let Let A_2, B_2, and C_2 be the points where L_2 intersects those side lines. The Droz-Farny line theorem says that the midpoints of the three segments A_1A_2, B_1B_2, and C_1C_2 are collinear. The theorem was stated by Arnold Droz-Farny in 1899, but it is not clear whether he had a proof. Goormaghtigh's generalization A generalization of the Droz-Farny line theorem was proved in 1930 by René Goormaghtigh. As above, let T be a triangle with vertices A, B, and C. Let P be any point distinct from A, B, and C, and L be any line through ...
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