Let ABC be a triangle. Prove that there is a unique point U in the plane of ABC such that there exist real numbers α,β,γ,δ not all zero, such that
\alpha PL^2 \plus{} \beta PM^2 \plus{} \gamma PN^2 \plus{} \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. geometry unsolvedgeometry