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 $PC$, respectively, 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. The generalization as follows:

First generalization: Let 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]

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. ^[7][8][9]

References

1. A. Droz-Farny (1899), "Question 14111". The Educational Times, volume 71, pages 89-90
2. Jean-Louis Ayme (2004), "A Purely Synthetic Proof of the Droz-Farny Line Theorem". Forum Geometricorum, volume 14, pages 219–224, ISSN 1534-1178
3. Floor van Lamoen and Eric W. Weisstein (), Droz-Farny Theorem at Mathworld
4. 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.
5. René Goormaghtigh (1930), "Sur une généralisation du théoreme de Noyer, Droz-Farny et Neuberg". Mathesis, volume 44, page 25
6. 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
7. 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, ISSN 2284-5569
8. Geoff Smith (2015). 99.20 A projective Simson line. The Mathematical Gazette, 99, pp 339-341. doi:10.1017/mag.2015.47
9. O.T.Dao 29-July-2013, Two Pascals merge into one, Cut-the-Knot