nLab Desargues' theorem

Idea

In most projective geometries, including all classical projective spaces, two distinct triangles are in a perspective with respect to a point iff they are in a perspective with respect to a line.

Statement

By a triangle we mean three noncolinear points.

In a projective space of any dimension greater than 2, in any projective plane over a field or skewfield, and in every projective plane where the Pappus theorem holds, the following is true. For any two triangles $A B C$ and $A'B'C'$ without common points the three lines $A A'$, $B B'$ and $C C'$ meet in a single point iff the intersections of pairs of sides $A B\cap A'B'$, $A C\cap A' C'$ and $B C\cap B'C'$ lie on a single line.

Nondesarguesian planes

In dimension 2, there exist nondesarguesian planes, both finite and infinite (in the sense of a number of points). All known finite nondesarguesian planes have the order which is a prime number to an integer power bigger than 1. The octonionic projective plane is an infinite example.

Desarguesian configuration

Desarguesian configuration consists of all 10 points and 10 lines mentioned in the statement, satisfying the condition of the perspective. Thus one includes 6 distinct points $A,B,C,A',B',C'$ such that $A,B,C$ are not colinear, $A',B',C'$ are not colinear and lines $A A'$, $B B'$ and $C C'$ meet in a single point $O$ which is the 7th point; then one adds the three intersections $A B\cap A'B'$, $A C\cap A' C'$ and $B C\cap B'C'$ which should be on the same line $p$. Lines are the three sides of each triangles, the lines $A A'$, $B B'$, $C C'$ and $p$. In higher dimension, the configuration may lay but does not need to lay on a single plane.

Related entries

• projective geometry

• projective plane

• octonionic projective plane

• harmonic ratio

References

• Emil Artin, §II.6 in: Geometric Algebra, Wiley 1957 (1988) [ISBN:978-1-118-16454-9, Wikipedia entry, ark:/13960/t4nk37034]

See also

• Wikipedia, Desargues’s theorem