ILL 1989 Problem 23

Let $ABC$ be a triangle. Prove that there is a unique point $U$ in the plane of $ABC$ such that there exist real numbers $\alpha, \beta, \gamma, \delta$ not all zero, such that

$$\alpha PL^2 + \beta PM^2 + \gamma PN^2 + \delta UP^2$$

is constant for all points $P$ of the plane, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively. Identify $U.$