One-point extensions of generalized hexagons and octagons

In this note, we prove the uniqueness of the one-point extension of a generalized hexagon of order 2 and prove the non-existence of such an extension of any other finite generalized hexagon of classical order, different from the one of order 2, and of the known finite generalized octagons provided the following property holds: for any three points x, y and z of , the graph-theoretic distance from y to z in the derived generalized hexagon is the same as the distance from x to z in .