## Desargues's theorem |

in

,projective geometry **desargues's theorem**, named after , states:girard desargues - two
are in perspectivetriangles *axially* they are inif and only if perspective *centrally*.

denote the three

of one triangle byvertices *a*,*b*and*c*, and those of the other by*a*,*b*and*c*.*axial*means that linesperspectivity *ab*and*ab*meet in a point, lines*ac*and*ac*meet in a second point, and lines*bc*and*bc*meet in a third point, and that these three points all lie on a common line called the*axis of perspectivity*.*central perspectivity*means that the three lines*aa*,*bb*and*cc*are concurrent, at a point called the*center of perspectivity*.this

is true in the usualintersection theorem but special care needs to be taken in exceptional cases, as when a pair of sides are parallel, so that their "point of intersection" recedes to infinity. commonly, to remove these exceptions, mathematicians "complete" the euclidean plane by adding points at infinity, followingeuclidean plane . this results in ajean-victor poncelet .projective plane desargues's theorem is true for the

, for any projective space defined arithmetically from areal projective plane orfield , for any projective space of dimension unequal to two, and for any projective space in whichdivision ring holds. however, there are manypappus's theorem in which desargues's theorem is false.planes - two
- history
- projective versus affine spaces
- self-duality
- proof of desargues's theorem
- relation to pappus's theorem
- the desargues configuration
- the little desargues theorem
- see also
- notes
- references
- external links

In **Desargues's theorem**, named after

- Two
triangles are in perspective*axially*if and only if they are inperspective *centrally*.

Denote the three *a*, *b* and *c*, and those of the other by *A*, *B* and *C*. *Axial perspectivity* means that lines

This

Desargues's theorem is true for the