Let
be a projective plane.
As usual, a quadrangle in
consists of four points (p_{1},p_{2},p_{3},
p_{4}), no three of which are on a common line. Given
such a quadrangle, we have the following three diagonal points.

The quadrangle is called a Fano quadrangle if the three diagonal
points d_{1},d_{2},d_{3} are not collinear.

The entire plane is a Fano plane if each quadrangle is a Fano quadrangle. Moreover, the plane is an Antifano plane if there are no Fano quadrangles in , i.e. if the diagonal points of any quadrangle are always collinear.

**Theorem.** Let V be a three dimensional vector space over a
skewfield K. Then the following holds.

- If the characteristic of K is distinct from 2, then the projective plane PG(V) is a Fano plane.
- On the other hand, if the characteristic of K equals 2, then PG(V) is an Antifano plane.

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