Desargues’ theorem

Let $ABC$ and $XYZ$ be two triangles. They are said to be perspective from a point if $AX,BY$ and $CZ$ are either concurrent or parallel, and are said to be perspective from a line if either the points of intersections $AB\cap XY,BC\cap YZ$ and $CA\cap ZX$ all exist and are collinear, or do not exist at all (three pairs of parallel lines).

Given two triangles such that no vertex of one triangle is the vertex of another:

• The Desarguesian property states: if they are perspective from a point, they are perspective from a line.

• The dual Desarguesian property states: if they are perspective from a line, they are perspective from a point.

A related concept is that of a Desarguesian configuration, which consists of two triangles which are both perspective from a point and perspective from a line. We say that the two triangles form a Desarguesian configuration. The point and the line are called the vertex and axis of the configuration. Note that the point may be a point at infinity, and the line may be a line at infinity. Below is a diagram of a Desarguesian configuration.

(XEukleides source codedesargues.euk for the drawing)

A geometry with points, lines and an incidence relation between them is said to be Desarguesian if, given any two triangles such that no vertex of one is the vertex of another, then both the Desarguesian property and its dual are true. Equivalently, a geometry is Desarguesian if whenever two triangles are in perspective from either a point or a line, then they form a Desarguesian configuration.

Desargues’ theorem. The Euclidean space is Desarguesian.

Remarks.

• In general, affine spaces and projective spaces are Desarguesian, provided that the space in question is at least dimension 3. If the dimension is 2, it can be shown that the space (affine or projective) is Desarguesian iff it can be embedded in a 3 dimensional space (affine or projective).

• In order to show that a projective space is Desarguesian, one only needs to show one of the two Desarguesian properties, since the other one may be automatically deduced according to the principal of duality. In addition, one may drop the case where two lines are parallel, as they always intersect at a point. In proving Desargues’ theorem, one generally “complete” the affine space into a projective one first, and use homogeneous coordinates to prove the theorem.

• A special type of Desarguesian configuration where its vertex lies on its axis is called a minor Desarguesian configuration. The minor Desarguesian property states that if two triangles are perspective from a point, then they form a minor Desarguesian configuration. Interchanging points and lines, we may form the dual minor Desarguesian property.

• Another special type of Desarguesian configuration occurs when the geometry is affine. A Desarguesian configuration is said to be affine if its axis is a line at infinity. In other words, given two triangles perspective from a point, they form a Desarguesian configuration in which their corresponding sides are parallel. An affine Desarguesian configuration is minor if its vertex is a point at infinity. In other words, the two triangles are such that not only are their corresponding sides parallel, the lines joining the corresponding vertices are parallel as well. These statements may also be dualized.

Title	Desargues’ theorem
Canonical name	DesarguesTheorem
Date of creation	2013-03-22 11:51:32
Last modified on	2013-03-22 11:51:32
Owner	drini (3)
Last modified by	drini (3)
Numerical id	19
Author	drini (3)
Entry type	Theorem
Classification	msc 51A30
Classification	msc 46L05
Defines	Desarguesian
Defines	dual Desarguesian
Defines	Desarguesian property
Defines	affine Desarguesian
Defines	minor Desarguesian
Defines	Desarguesian configuration
Defines	minor affine Desarguesian