Droz-Farny line theorem

Short description: Property of perpendicular lines through orthocenters

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 $P{A}_1$, $P{B}_1$, and $P{C}_1$ 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.