# 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 \PMlinktofilesource 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.

Remarks.

 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