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.

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

Let ${\displaystyle T}$ be a triangle with vertices ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$, and let ${\displaystyle H}$ be its orthocenter (the common point of its three altitude lines. Let ${\displaystyle L_{1}}$ and ${\displaystyle L_{2}}$ be any two mutually perpendicular lines through ${\displaystyle H}$. Let ${\displaystyle A_{1}}$, ${\displaystyle B_{1}}$, and ${\displaystyle C_{1}}$ be the points where ${\displaystyle L_{1}}$ intersects the side lines ${\displaystyle BC}$, ${\displaystyle CA}$, and ${\displaystyle AB}$, respectively. Similarly, let Let ${\displaystyle A_{2}}$, ${\displaystyle B_{2}}$, and ${\displaystyle C_{2}}$ be the points where ${\displaystyle L_{2}}$ intersects those side lines. The Droz-Farny line theorem says that the midpoints of the three segments ${\displaystyle A_{1}A_{2}}$, ${\displaystyle B_{1}B_{2}}$, and ${\displaystyle C_{1}C_{2}}$ are collinear.^[1][2][3]

The theorem was stated by Arnold Droz-Farny in 1899,^[1] but it is not clear whether he had a proof.^[4]

Goormaghtigh's generalization

A generalization of the Droz-Farny line theorem was proved in 1930 by René Goormaghtigh.^[5]

As above, let ${\displaystyle T}$ be a triangle with vertices ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$. Let ${\displaystyle P}$ be any point distinct from ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$, and ${\displaystyle L}$ be any line through ${\displaystyle P}$. Let ${\displaystyle A_{1}}$, ${\displaystyle B_{1}}$, and ${\displaystyle C_{1}}$ be points on the side lines ${\displaystyle BC}$, ${\displaystyle CA}$, and ${\displaystyle AB}$, respectively, such that the lines ${\displaystyle PA_{1}}$, ${\displaystyle PB_{1}}$, and ${\displaystyle PC_{1}}$ are the images of the lines ${\displaystyle PA}$, ${\displaystyle PB}$, and ${\displaystyle PC}$, respectively, by reflection against the line ${\displaystyle L}$. Goormaghtigh's theorem then says that the points ${\displaystyle A_{1}}$, ${\displaystyle B_{1}}$, and ${\displaystyle C_{1}}$ are collinear.

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

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.^[6]

Dao's first generalization

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. ^[7][8][9]