BG-MO 2013 Problem 5

Consider acute $\triangle ABC$ with altitudes $AA_1, BB_1$ and $CC_1$ ( $A_1 \in BC,B_1 \in AC,C_1 \in AB$ ). A point $C'$ on the extension of $B_1A_1$ beyond $A_1$ is such that $A_1C' = B_1C_1$ . Analogously, a point $B'$ on the extension of A $_1C_1$ beyond $C_1$ is such that $C_1B' = A_1B_1$ and a point $A'$ on the extension of $C_1B_1$ beyond $B_1$ is such that $B_1A' = C_1A_1$ . Denote by $A'', B'', C''$ the symmetric points of $A' , B' , C'$ with respect to $BC, CA$ and $AB$ respectively. Prove that if $R, R'$ and R'' are circumradiii of $\triangle ABC, \triangle A'B'C'$ and $\triangle A''B''C''$ , then $R, R'$ and $R''$ are sidelengths of a triangle with area equals one half of the area of $\triangle ABC$ .