Desargues' Theorem

Desargues' theorem in Euclidean geometry.

Desargues' theorem says that if two triangles are in perspective from a point, then they're also in perspective from a line. In the diagram below, the red triangle ABC and the green triangle DEF are in perspective from the point P. That means pairs of corresponding points on the triangles are collinear with P, so that ADP, BEP, and CFP are three straight lines. (They're drawn in black.) Desargues discovered that the three corresponding sides of the two triangles meet at three collinear points. Sides AB and DE meet at point G, Sides BC and EF meet at point H, and sides CA and FD meet at point K, and these three points GHK lie on a line. When the three corresponding sides of two triangles meet at collinear points, the triangle is said to be in perspective from a line.

Desargues Theorem in Euclidean Geometry

The argument for the validity of Desargue's theorem is easy if P and the two triangles ABC and DEF are not all in the same plane. In that case, triangle ABC is in one plane, the red plane, while triangle DEF is in another plane, the green plane. The red and green planes meet in a line, the blue line, and the points G, H, and K have to lie on this line since all three of them lie in both the red and green planes. In order to prove the case where P and the two triangles do lie in the same plane, just lift the triangles into the spatial case, and the projection back to the original plane gives the result.

Desargues' theorem in round geometry.

Desargues' theorem holds in round geometry as well. For that, ABC and DEF are each two round triangles with dual triangles A*B*C* and D*E*F*, respectively, and P a point on each of the circles ADA*D*, BEB*E*, and CFC*F*. In other words, we can say the triangles are in perspective from the point P. Then the two triangles are also in perspective from a line. That is, sides ABA*B* and DED*E* meet at G and G*, sides BCB*C* and EFE*F* meet at H and H*, and sides CAC*A* and FDF*D* meet at K and K*, and the points GHKG*H*K* lie on a circle.

Desargues Theorem in Round Geometry

Desargues' theorem in Hyperbolic Geometry

The version of Desargue's theorem in Round Geometry specializes to one in hyperbolic geometry, and also one in Elliptic geometry.

Desargues Theorem in Hyperbolic Geometry

Desargues' theorem in Elliptic Geometry

Desargues Theorem in Elliptic Geometry

David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610