Clash of circles!

[Kazi_Zareer]

In $\triangle ABC$ Let $W_a$ be the circle that passes throw $B,C $ and it is tangent to the incircle of $\triangle ABC$. We define $W_b,W_c$ similarly. Let $W_a\cap W_b=B', W_a\cap W_c=C'$ let $BC'\cap AC=B'', CB'\cap AB=C''$ let $I$ be the incenter of $\triangle ABC$. Prove that $B''$,$I$, $C''$ are collinear if and if only $AB+AC=3BC$.