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 andOR 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, andEquating (1) and (2) we haveHence the ratio is independent of α, and thereforethe angle α. being in circular measure.