IR-TST 2012 Problem 6

The pentagon $ABCDE$ is inscirbed in a circle $w$ . Suppose that $w_a,w_b,w_c,w_d,w_e$ are reflections of $w$ with respect to sides $AB,BC,CD,DE,EA$ respectively. Let $A'$ be the second intersection point of $w_a,w_e$ and define $B',C',D',E'$ similarly. Prove that $$2\le \frac{S_{A'B'C'D'E'}}{S_{ABCDE}}\le 3,$$ where $S_X$ denotes the surface of figure $X$ .

Proposed by Morteza Saghafian, Ali khezeli