DesarguesTheorem 2001-10-20 22:37:07 2006-07-29 14:55:14 Theorem \usepackage{graphicx} %\usepackage{xypic} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand{\mathbb}[1]{\mathbbmss{#1}} \newcommand{\figura}[1]{\begin{center}\includegraphics{#1}\end{center}} \newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}} If $ABC$ and $XYZ$ are two triangles in perspective (that is, $AX,BY$ and $CZ$ are concurrent or parallel) then the points of intersection of the three pairs of lines $(BC,YZ), (CA,ZX), (AB,XY)$ are collinear. Also, if $ABC$ and $XYZ$ are triangles with distinct vertices and the intersection of $BC$ with $YZ$, the intersection of $CA$ with $ZX$ and the intersection of $AB$ with $XY$ are three collinear points, then the triangles are in perspective. \figuraex{desargues}{scale=0.75} {\footnotesize(XEukleides \PMlinktofile{source code}{desargues.euk} for the drawing)}