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.
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 da, db, dc 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' ∩ BC =A0; DB' ∩ AC = B0; DC' ∩ AB= C0. Then A0, B0, C0 are collinear. 
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