On strict strong constructibility with a compass alone

We show that every point in the plane which can be constructed by a compass and a ruler, given a set S of points, can be constructed using a compass alone so that the following restriction is met. Let O and K be two arbitrarily chosen distinct points of S. Then every point is obtained as a proper intersection of two circles that are either completely symmetrical with respect to the line OK or have both their centers on this line.