Problems

Prove that the area $S_{ABC}$ of a triangle is equal to $abc/4R$.

The point $D$ lies on the base $AC$ of the isosceles triangle $ABC$. Prove that the radii of the circumscribed circles of the triangles $ABD$ and $CBD$ are equal.

Express the area of the triangle $ABC$ through the length of the side $BC$ and the angles $B$ and $C$.