Desargues' Theorem

This theorem was discovered by the French mathematician Girard Desargues in 1639. It motivated the development, in the first quarter of the 19th century, of projective geometry by another French mathematician, Jean-Victor Poncelet. The theorem states that if two triangles, and , situated in three-dimensional space, are related to each other in such a way that they can be seen perspectively from one point (i.e., the lines Aa, Bb, and Cc all intersect in one point), then the points of intersection of corresponding sides all lie on one line, provided that no two corresponding sides are parallel. Should this last case occur, there will be only two points of intersection instead of three, and the theorem must be modified to include the result that these two points will lie on a line parallel to the two parallel sides of the triangles. Rather than modify the theorem to cover this special case, Poncelet instead modified Euclidean space itself by postulating points at infinity, which was the key for the development of projective geometry. In this new projective space (Euclidean space with added points at infinity), each straight line is given an added point at infinity, with parallel lines having a common point. Slightly edited from: Britannica, The Editors of Encyclopaedia. "Desargues's theorem". Encyclopedia Britannica, 20 Jul. 1998, Accessed 15 April 2021.