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 $$P$$. Let $$A_1$$, $$B_1$$, and $$C_1$$ be points on the side lines $$BC$$, $$CA$$, and $$AB$$, respectively, such that the lines $$PA_1$$, $$PB_1$$, and $$PC_1$$ are the images of the lines $$PA$$, $$PB$$, and $$PC$$, 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 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.