In Euclidean geometry, the Droz-Farny line theorem is a property of two perpendicular lines through the orthocenter of an arbitrary triangle.
Let be a triangle with vertices , , and , and let be its orthocenter (the common point of its three altitude lines. Let and be any two mutually perpendicular lines through . Let , , and be the points where intersects the side lines , , and , respectively. Similarly, let Let , , and be the points where intersects those side lines. The Droz-Farny line theorem says that the midpoints of the three segments , , and are collinear.
The theorem was stated by Arnold Droz-Farny in 1899, but it is not clear whether he had a proof.
A generalization of the Droz-Farny line theorem was proved in 1930 by René Goormaghtigh.
As above, let be a triangle with vertices , , and . Let be any point distinct from , , and , and be any line through . Let , , and be points on the side lines , , and , respectively, such that the lines , , and are the images of the lines , , and , respectively, by reflection against the line . Goormaghtigh's theorem then says that the points , , and are collinear.
The Droz-Farny line theorem is a special case of this result, when is the orthocenter of triangle .
The theorem was further generalized by Dao Thanh Oai. The generalization as follows:
First generalization: Let ABC be a triangle, P be a point on the plane, let three parallel segments AA', BB', CC' such that its midpoints and P are collinear. Then PA', PB', PC' meet BC, CA, AB respectively at three collinear points.
Second generalization: Let a conic S and a point P on the plane. Construct three lines d<sub>a</sub>, d<sub>b</sub>, d<sub>c</sub> through P such that they meet the conic at A, A'; B, B' ; C, C' respectively. Let D be a point on the polar of point P with respect to (S) or D lies on the conic (S). Let DA' â© BC =A<sub>0</sub>; DB' â© AC = B<sub>0</sub>; DC' â© AB= C<sub>0</sub>. Then A<sub>0</sub>, B<sub>0</sub>, C<sub>0</sub> are collinear.