## 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.
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

Clark University

Worcester, MA 01610

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