KöMaL 2013 B4530

Prove that the midpoint of the line segment connecting the centres of two regular polygons of the same number of sides drawn on the sides $AC$ and $BC$ of a triangle ABC on the outside coincides with the centre of the regular polygon of the same number of sides drawn on the line segment connecting the midpoints of the sides $AC$and $BC$, on the side where the vertex $C$ lies.