In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, the Droz-Farny line theorem is a property of two perpendicular lines through the
orthocenter
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...
of an arbitrary triangle.
Let
be a triangle with vertices
,
, and
, and let
be its orthocenter (the common point of its three
altitude lines. Let
and
be any two mutually perpendicular lines through
. Let
,
, and
be the points where
intersects the side lines
,
, and
, respectively. Similarly, let Let
,
, and
be the points where
intersects those side lines. The Droz-Farny line theorem says that the midpoints of the three segments
,
, and
are
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
.
[
The theorem was stated by ]Arnold Droz-Farny
Arnold Droz-Farny (12 February 1856 – 14 January 1912) was a Swiss mathematician, professor in High School of Porrentruy (near Basel).
Life and work
Arnold Droz changed his family name later in his life when he married Lisa Farny. He studied ...
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 be a triangle with vertices , , and . Let be any point distinct from , , and , and be any line through . Let , , and be points on the side lines , , and , respectively, such that the lines , , and are the images of the lines , , and , respectively, by reflection against the line . Goormaghtigh's theorem then says that the points , , and are collinear.
The Droz-Farny line theorem is a special case of this result, when is the orthocenter of triangle .
]
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.[
Second generalization: Let a ]conic
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
S and a point P on the plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
. Construct three lines
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
da, db, dc 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),]
A Purely Synthetic Proof of the Droz-Farny Line Theorem
. ''Forum Geometricorum'', volume 14, pages 219–224,
[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,
[Floor van Lamoen and Eric W. Weisstein (), ]
' at Mathworld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...
[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), ]
Arnold Droz-Farny
'. The MacTutor History of Mathematics archive. Online document, accessed on 2014-10-05.
[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]
/ref>
[O.T.Dao 29-July-2013]
Two Pascals merge into one
Cut-the-Knot
Euclidean geometry
Conic sections
Theorems about triangles