Let
be a projective plane.
Two triangles D_{i}=(a_{i},b_{i},c_{i})
are central to a point z if
,
,
and
,
,
.

Dually, the two triangles are axial to a line a if for i=1,2, , , and , , , .

The plane
is (a,z)-desarguesian if given any pair of triangles D_{i}=
(a_{i},b_{i},c_{i}) (i=1,2) central to z with

the two triangles D_{1} and D_{2} are already axial
with respect to a.

Finally, is desarguesian if is (a,z)-desarguesian for all points z and all lines a.

**Theorem.** Let
be a projective plane,
.
Then
is (a,z)-desarguesian if and only if
is (a,z)-transitive.

**Theorem.** A projective plane
is desarguesian if and only if there exists a three-dimensional
vector space V over a skewfield K such that
where PG(V) denotes the projective plane
over V.

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