VN-TST 1994 Problem 1

Given a parallelogram $ABCD$ . Let $E$ be a point on the side $BC$ and $F$ be a point on the side $CD$ such that the triangles $ABE$ and $BCF$ have the same area. The diaogonal $BD$ intersects $AE$ at $M$ and intersects $AF$ at $N$ . Prove that:

I. There exists a triangle, three sides of which are equal to $BM, MN, ND$ . II. When $E, F$ vary such that the length of $MN$ decreases, the radius of the circumcircle of the above mentioned triangle also decreases.