VN-TST 1997 Problem 1

Let $ABCD$ be a given tetrahedron, with $BC = a$ , $CA = b$ , $AB = c$ , $DA = a_1$ , $DB = b_1$ , $DC = c_1$ . Prove that there is a unique point $P$ satisfying $$PA^2 + a_1^2 + b^2 + c^2 = PB^2 + b_1^2 + c^2 + a^2 = PC^2 + c_1^2 + a^2 + b^2 = PD^2 + a_1^2 + b_1^2 + c_1^2$$ and for this point $P$ we have $PA^2 + PB^2 + PC^2 + PD^2 \ge 4R^2$ , where $R$ is the circumradius of the tetrahedron $ABCD$ . Find the necessary and sufficient condition so that this inequality is an equality.