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.

be a triangle with vertices

be its orthocenter (the common point of its three altitude lines.

be any two mutually perpendicular lines through

intersects the side lines

{\displaystyle BC}

Similarly, let Let

intersects those side lines.

The Droz-Farny line theorem says that the midpoints of the three segments

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

be a triangle with vertices

be any point distinct from

be any line through

be points on the side lines

{\displaystyle CA}

, respectively, such that the lines

are the images of the lines

, respectively, by reflection against the line

Goormaghtigh's theorem then says that the points

The Droz-Farny line theorem is a special case of this result, when

is the orthocenter of triangle

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] 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.