Droz-Farny line theorem

The line through ${\displaystyle A_{0},B_{0},C_{0}}$ 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 ${\displaystyle T}$ be a triangle with vertices ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$, and let ${\displaystyle H}$ be its orthocenter (the common point of its three altitude lines. Let ${\displaystyle L_{1}}$ and ${\displaystyle L_{2}}$ be any two mutually perpendicular lines through ${\displaystyle H}$. Let ${\displaystyle A_{1}}$, ${\displaystyle B_{1}}$, and ${\displaystyle C_{1}}$ be the points where ${\displaystyle L_{1}}$ intersects the side lines ${\displaystyle BC}$, ${\displaystyle CA}$, and ${\displaystyle AB}$, respectively. Similarly, let Let ${\displaystyle A_{2}}$, ${\displaystyle B_{2}}$, and ${\displaystyle C_{2}}$ be the points where ${\displaystyle L_{2}}$ intersects those side lines. The Droz-Farny line theorem says that the midpoints of the three segments ${\displaystyle A_{1}A_{2}}$, ${\displaystyle B_{1}B_{2}}$, and ${\displaystyle C_{1}C_{2}}$ 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 ${\displaystyle T}$ be a triangle with vertices ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$. Let ${\displaystyle P}$ be any point distinct from ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$, and ${\displaystyle L}$ be any line through ${\displaystyle P}$. Let ${\displaystyle A_{1}}$, ${\displaystyle B_{1}}$, and ${\displaystyle C_{1}}$ be points on the side lines ${\displaystyle BC}$, ${\displaystyle CA}$, and ${\displaystyle AB}$, respectively, such that the lines ${\displaystyle PA_{1}}$, ${\displaystyle PB_{1}}$, and ${\displaystyle PC_{1}}$ are the images of the lines ${\displaystyle PA}$, ${\displaystyle PB}$, and ${\displaystyle PC}$, respectively, by reflection against the line ${\displaystyle L}$. Goormaghtigh's theorem then says that the points ${\displaystyle A_{1}}$, ${\displaystyle B_{1}}$, and ${\displaystyle C_{1}}$ are collinear.

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

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. 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 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
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 Archived 2014-10-06 at the Wayback Machine, 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