In Euclidean geometry, the **Droz-Farny line theorem** is a property of two perpendicular lines through the orthocenter of an arbitrary triangle.

Let be a triangle with vertices , , and , and let be its orthocenter (the common point of its three altitude lines. Let and be any two mutually perpendicular lines through . Let , , and be the points where intersects the side lines , , and , respectively. Similarly, let Let , , and be the points where intersects those side lines. The Droz-Farny line theorem says that the midpoints of the three segments , , and 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 be a triangle with vertices , , and . Let be any point distinct from , , and , and be any line through . Let , , and be points on the side lines , , and , respectively, such that the lines , , and are the images of the lines , , and , respectively, by reflection against the line . Goormaghtigh's theorem then says that the points , , and are collinear.

The Droz-Farny line theorem is a special case of this result, when is the orthocenter of triangle .

## 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]}

**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 =A_{0}; DB' ∩ AC = B_{0}; DC' ∩ AB= C_{0}. Then A_{0}, B_{0}, C_{0} are collinear. ^{[7]}^{[8]}^{[9]}

## References

- ^
^{a}^{b}A. Droz-Farny (1899), "Question 14111".*The Educational Times*, volume 71, pages 89-90 **^**Jean-Louis Ayme (2004), "A Purely Synthetic Proof of the Droz-Farny Line Theorem".*Forum Geometricorum*, volume 14, pages 219–224, ISSN 1534-1178**^**Floor van Lamoen and Eric W. Weisstein (),*Droz-Farny Theorem*at Mathworld**^**J. J. O'Connor and E. F. Robertson (2006),*Arnold Droz-Farny*. The MacTutor History of Mathematics archive. Online document, accessed on 2014-10-05.**^**René Goormaghtigh (1930), "Sur une généralisation du théoreme de Noyer, Droz-Farny et Neuberg".*Mathesis*, volume 44, page 25**^**Son Tran Hoang (2014), "A synthetic proof of Dao's generalization of Goormaghtigh's theorem Archived 2014-10-06 at the Wayback Machine."*Global Journal of Advanced Research on Classical and Modern Geometries*, volume 3, pages 125–129, ISSN 2284-5569**^**Nguyen Ngoc Giang,*A proof of Dao theorem*, Global Journal of Advanced Research on Classical and Modern Geometries, Vol.4, (2015), Issue 2, page 102-105 Archived 2014-10-06 at the Wayback Machine, ISSN 2284-5569**^**Geoff Smith (2015).*99.20 A projective Simson line*. The Mathematical Gazette, 99, pp 339-341. doi:10.1017/mag.2015.47**^**O.T.Dao 29-July-2013, Two Pascals merge into one, Cut-the-Knot