Školjka

Let $ABC$ be a triangle with $|AB|=|AC|$. Also, let $D\in \overline{BC}$ be a point such that $|BC|>|BD|>|DC|>0$, and let $\mathcal{C}_1,\mathcal{C}_2$ be the circumcircles of the triangles $ABD$ and $ADC$ respectively. Let $\overline{BB'}$ and $\overline{CC'}$ be diameters in the two circles, and let $M$ be the midpoint of $\overline{B'C'}$. Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$).