Moufang plane

Remarks

So three collinear intersection points are required only when the centre $O$ lies on the axis $\ell$. Equivalently, every line of the plane is a translation line, that is, the group of elations with axis $\ell$ acts transitively on the points off $\ell$. A projective plane is a Moufang plane exactly when it can be coordinatized by an alternative division ring.

Example: $\mathbb{P}^2(k)$ over a field $k$ is a Moufang plane. Check: Desargues's theorem holds in $\mathbb{P}^2(k)$ for all triangles in perspective, in particular when the centre lies on the axis.

By the Artin–Zorn theorem every finite alternative division ring is a field, so every finite Moufang plane is Desarguesian.

Non-example: the Hall plane of order $9$ is a finite non-Desarguesian plane. So by the previous sentence it is not a Moufang plane: the little Desargues property fails in it for some line $\ell$ and point $O$.