Why do the intersections of corresponding sides in Desarguesâs theorem lie on one line?
The sight lines drawn from the image in the reality plane (RP) to the artist's eye intersect the picture plane (PP) to form a projective, or perspective, drawing. The horizontal line drawn parallel to PP corresponds to the horizon. Early perspective experimenters sometimes used translucent paper or glass for the picture plane, which they drew on while looking through a small hole to keep their focus steady.
Desarguesâs theorem states that if two triangles are in perspective from a pointâmeaning the lines connecting their corresponding vertices all meet at a single pointâthen the points where their corresponding sides intersect are collinear (they lie on a single line).1
In the context of projective geometry, this result is a consequence of the properties of projections. When triangles are viewed from a central point in three-dimensional space, the lines forming the sides of the triangles lie in planes that intersect at the line containing the intersection points of the corresponding sides.1
By postulating points at infinity, mathematicians like Jean-Victor Poncelet were able to treat parallel lines as lines that intersect at a common point at infinity. This framework allows the theorem to hold universally, as collinearity is an invariant property preserved under projective mappings. Essentially, the intersection points are forced onto a single line because they represent the intersection of the planes defined by the two triangles.123