JBMO 1999 Problem 4

Let $ABC$ be a triangle with $AB=AC$ . Also, let $D\in[BC]$ 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 $BB'$ and $CC'$ be diameters in the two circles, and let $M$ be the midpoint of $B'C'$ . Prove that the area of the triangle $MBC$ is constant (i.e. it does not depend on the choice of the point $D$ ).

Greece