Let AB and BC be two circular arcs subtending angles 2α and 2β at the common centre 0. From symmetry the centroids G1, G2 and G of AB, BC and AC lie on the bisectors OP, OQ and

OR of the angles which they subtend at the centre. Also, G is the centroid of two particles placed at G1 and G2, and with masses proportional to the arcs AB and BC. Hence G1, G, and G2 are collinear, and

Equating (1) and (2) we have

Hence the ratio is independent of α, and therefore

the angle α. being in circular measure.