Find two models of incidence geometry each with exactly four

Find two models of incidence geometry, each with exactly four points, that are not isomorphic. Justify your answer.

Solution

Six lines with each line having two points. It has the Euclidean parallel property that states, for every line l and for every point P lying not on l, there exists a unique parallel m to l through point P.

Second model has four lines, one of which has three points. It has the elliptic parallel property that states incidence geometry has the elliptic parallel property iff any two lines do intersect necessarily at a unique point.