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 ${\displaystyle T}$ .

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 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]