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 of the three segments $A_{1}A_{2}$ , $B_{1}B_{2}$ , and $C_{1}C_{2}$ 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 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]

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 2 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
↑ 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
↑ 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

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