Some notes on generalized quadrangles of order s with a span of regular points

Let Q be a generalized quadrangle of order s with a regular point x. The set x⊥ together with all spans which are contained in x⊥ define a projective plane of order s. We introduce a property (Py) for every point y of Q noncollinear with x and prove that this property is equivalent with the regularity of the point y. We will use this to give an elementary proof for the following result: every generalized quadrangle of order q which has a center of symmetry x such that πx ∼= PG(2, q) and a regular point y not collinear with x is isomorphic to W (q).