Droz-Farny line theorem

The line through

A_0,B_0,C_0

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

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

Dao's generalization

The theorem was further generalized by Dao Thanh Oai.^[6]

References

↑ 1.0 1.1 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." Global Journal of Advanced Research on Classical and Modern Geometries, volume 3, pages 125–129, ISSN 2284-5569