Droz Farny line theorem

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 P . Let A 1 , B 1 , and C 1 be points on the side lines B C , C A , and A B , respectively, such that the lines P A 1 , P B 1 , and P C 1 are the images of the lines P A , P B , and P C , respectively, by reflection against the line L . Goormaghtigh's theorem then says that the points A 1 , B 1 , and C 1 are collinear.

The Droz-Farny line theorem is a special case of this result, when P is the orthocenter of triangle T .

Dao's generalization

The theorem was further generalized by Dao Thanh Oai. The generalization as follows:

First generalization: Let 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_a, d_b, d_c 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.