Let ABC be a triangle with AB=AC. Also, let D∈[BC] be a point such that BC>BD>DC>0, and let C1,C2 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 geometrycircumcircletrigonometryperpendicular bisectorJBMO