Droz-Farny line theorem

Short description: Property of perpendicular lines through orthocenters

The line through [math]\displaystyle{ A_0,B_0,C_0 }[/math] 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 [math]\displaystyle{ T }[/math] be a triangle with vertices [math]\displaystyle{ A }[/math], [math]\displaystyle{ B }[/math], and [math]\displaystyle{ C }[/math], and let [math]\displaystyle{ H }[/math] be its orthocenter (the common point of its three altitude lines. Let [math]\displaystyle{ L_1 }[/math] and [math]\displaystyle{ L_2 }[/math] be any two mutually perpendicular lines through [math]\displaystyle{ H }[/math]. Let [math]\displaystyle{ A_1 }[/math], [math]\displaystyle{ B_1 }[/math], and [math]\displaystyle{ C_1 }[/math] be the points where [math]\displaystyle{ L_1 }[/math] intersects the side lines [math]\displaystyle{ BC }[/math], [math]\displaystyle{ CA }[/math], and [math]\displaystyle{ AB }[/math], respectively. Similarly, let Let [math]\displaystyle{ A_2 }[/math], [math]\displaystyle{ B_2 }[/math], and [math]\displaystyle{ C_2 }[/math] be the points where [math]\displaystyle{ L_2 }[/math] intersects those side lines. The Droz-Farny line theorem says that the midpoints of the three segments [math]\displaystyle{ A_1A_2 }[/math], [math]\displaystyle{ B_1B_2 }[/math], and [math]\displaystyle{ C_1C_2 }[/math] 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 [math]\displaystyle{ T }[/math] be a triangle with vertices [math]\displaystyle{ A }[/math], [math]\displaystyle{ B }[/math], and [math]\displaystyle{ C }[/math]. Let [math]\displaystyle{ P }[/math] be any point distinct from [math]\displaystyle{ A }[/math], [math]\displaystyle{ B }[/math], and [math]\displaystyle{ C }[/math], and [math]\displaystyle{ L }[/math] be any line through [math]\displaystyle{ P }[/math]. Let [math]\displaystyle{ A_1 }[/math], [math]\displaystyle{ B_1 }[/math], and [math]\displaystyle{ C_1 }[/math] be points on the side lines [math]\displaystyle{ BC }[/math], [math]\displaystyle{ CA }[/math], and [math]\displaystyle{ AB }[/math], respectively, such that the lines [math]\displaystyle{ PA_1 }[/math], [math]\displaystyle{ PB_1 }[/math], and [math]\displaystyle{ PC_1 }[/math] are the images of the lines [math]\displaystyle{ PA }[/math], [math]\displaystyle{ PB }[/math], and [math]\displaystyle{ PC }[/math], respectively, by reflection against the line [math]\displaystyle{ L }[/math]. Goormaghtigh's theorem then says that the points [math]\displaystyle{ A_1 }[/math], [math]\displaystyle{ B_1 }[/math], and [math]\displaystyle{ C_1 }[/math] are collinear.

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

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.