Let be a projective plane. As usual, a quadrangle in consists of four points (p1,p2,p3, p4), 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 d1,d2,d3 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.

Note that there exist projective planes which are neither Fano planes nore Antifano planes.
Contributed by Hauke Klein
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