Desargues' theorem
/ day-ZARG /
Two triangles are said to be in perspective from a point when the three lines joining their matching corners all pass through one common point — picture two triangular shadows cast by a single lamp, their corresponding vertices lined up along rays from the bulb. Desargues' theorem says something remarkable then happens on the other side of the figure: the three pairs of matching sides, extended, meet in three points that are collinear — they all lie on one straight line. Perspective from a point forces perspective from a line.
Precisely. Let triangles ABC and A'B'C' be such that lines AA', BB', CC' all pass through a single point O (the center of perspectivity). Consider the three pairs of corresponding sides: BC with B'C', CA with C'A', AB with A'B'. Each pair, prolonged, meets in a point — call them P, Q, R. Desargues' theorem asserts P, Q, R are collinear; the line through them is the axis of perspectivity. The converse is also true: if the three pairs of corresponding sides meet in three collinear points, then the three vertex-joining lines are concurrent. In the projective plane there are no exceptions to dodge — if two corresponding sides happen to be 'parallel', their meeting point simply lands on the line at infinity, and the theorem still reads true.
Desargues' theorem (Girard Desargues, 1639) is a cornerstone of projective geometry and a perfect showcase of duality: the theorem and its converse are exactly dual statements, so the configuration is self-dual — ten points and ten lines with each point on three lines and each line through three points. A deep and honest subtlety: in the plane Desargues' theorem cannot be proved from the incidence axioms alone (there exist strange 'non-Desarguesian' planes where it fails), but it becomes automatic once the plane sits inside three-dimensional space — its natural proof lifts the two triangles into different planes meeting along the axis. That a plane fact is cleanest when viewed from space is one of the loveliest lessons in geometry.
Hold a small triangular card between a lamp and a wall, tilted, so its shadow is a second, differently shaped triangle. The lamp, the card's corner, and the shadow's matching corner are collinear for all three corners — that is perspective from the lamp. Desargues guarantees that where each card-edge's line crosses its shadow-edge's line, the three crossing points line up straight: perspective from a line.
Perspective from a point (three vertex-lines concurrent) forces perspective from a line (three side-intersections collinear).
Honest subtlety: Desargues' theorem is NOT a consequence of the plane incidence axioms alone — non-Desarguesian projective planes exist. It holds in every projective plane that can be embedded in a projective 3-space (in particular the real plane), where the easy proof comes from lifting into the third dimension.