On equi-angular points

A point $T$ is an equi-angular point of a collection of localized vectors if all of them are seen from $T$ at an equal oriented angle. It is proved that almost all triples of vectors in the plane with fixed origins (not all of which coincide) have an euqi-angular point. As a consequence, it is proved that if a triple of vectors in the plane has no equi-angular point, then their projections to a certain axis are equal.