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.
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 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 Elliptic Geometry
David E. Joyce
Department of Mathematics and Computer Science
Worcester, MA 01610