Theorems on 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, in such a way that the centres of all the circles used are on one particular segment OK, where O and K are two arbitrarily chosen distinct points of S. This strengthens (and at the same time also gives an alternative simple proof to) a famous theorem of Mascheroni and Mohr.