ILL 1971 Problem 6

Let squares be constructed on the sides $BC,CA,AB$ of a triangle $ABC$ , all to the outside of the triangle, and let $A_1,B_1, C_1$ be their centers. Starting from the triangle $A_1B_1C_1$ one analogously obtains a triangle $A_2B_2C_2$ . If $S, S_1, S_2$ denote the areas of triangles $ABC,A_1B_1C_1,A_2B_2C_2$ , respectively, prove that $S = 8S_1 - 4S_2.$