Droz-Farny line theorem

The line through $A_{0},B_{0},C_{0}$ is Droz-Farny line

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 $A_{1}A_{2}$, $B_{1}B_{2}$, and $C_{1}C_{2}$ 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.

Dao's second generalization

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.