Droz-Farny line theorem

{{short description|Property of perpendicular lines through orthocenters}}

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 $A_1A_2$ , $B_1B_2$ , and $C_1C_2$ are collinear.

The theorem was stated by Arnold Droz-Farny in 1899, but it is not clear whether he had a proof.

Goormaghtigh's generalization

A generalization of the Droz-Farny line theorem was proved in 1930 by René Goormaghtigh.

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.

File:Dao_generalization.svg

File:Daotheoremonconic1.svg

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.

References

Jean-Louis Ayme (2004), "[http://forumgeom.fau.edu/FG2004volume4/FG200426index.html A Purely Synthetic Proof of the Droz-Farny Line Theorem]". Forum Geometricorum, volume 14, pages 219–224, {{ISSN|1534-1178}}

Son Tran Hoang (2014), "[http://gjarcmg.geometry-math-journal.ro/ A synthetic proof of Dao's generalization of Goormaghtigh's theorem] {{Webarchive|url=https://web.archive.org/web/20141006193110/http://gjarcmg.geometry-math-journal.ro/ |date=2014-10-06 }}." Global Journal of Advanced Research on Classical and Modern Geometries, volume 3, pages 125–129, {{ISSN|2284-5569}}

Floor van Lamoen and Eric W. Weisstein (), [http://mathworld.wolfram.com/Droz-FarnyTheorem.html Droz-Farny Theorem] at Mathworld

A. Droz-Farny (1899), "Question 14111". The Educational Times, volume 71, pages 89-90

René Goormaghtigh (1930), "Sur une généralisation du théoreme de Noyer, Droz-Farny et Neuberg". Mathesis, volume 44, page 25

J. J. O'Connor and E. F. Robertson (2006), [http://www-history.mcs.st-and.ac.uk/Biographies/Droz-Farny.html Arnold Droz-Farny]. The MacTutor History of Mathematics archive. Online document, accessed on 2014-10-05.

[http://gjarcmg.geometry-math-journal.ro/ 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] {{Webarchive|url=https://web.archive.org/web/20141006193110/http://gjarcmg.geometry-math-journal.ro/ |date=2014-10-06 }}, {{ISSN|2284-5569}}

[http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9834854&fileId=S0025557215020549 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, [http://www.cut-the-knot.org/m/Geometry/DoublePascalConic.shtml Two Pascals merge into one], Cut-the-Knot

Category:Euclidean geometry

Category:Conic sections

Category:Theorems about triangles