ILL 1989 Problem 62

Given a convex polygon $A_1A_2 \ldots A_n$ with area $S$ and a point $M$ in the same plane, determine the area of polygon $M_1M_2 \ldots M_n,$ where $M_i$ is the image of $M$ under rotation $R^{\alpha}_{A_i}$ around $A_i$ by $\alpha_i, i = 1, 2, \ldots, n.$