Desargues' theorem

Desargues' theorem is a key idea in projective geometry stating that if two triangles are positioned so that lines connecting their corresponding vertices all meet at a single point (called perspective from a point), then the intersections of their corresponding sides will all lie on a single straight line (called perspective from a line). Conversely, if the sides of the triangles intersect along a common line, then the lines connecting the corresponding vertices will meet at a single point. This theorem reveals a deep relationship between how triangles are positioned and their internal and external intersections.