TR-MO 1995 Problem 2

Let $ABC$ be an acute triangle and let $k_{1},k_{2},k_{3}$ be the circles with diameters $BC,CA,AB$ , respectively. Let $K$ be the radical center of these circles. Segments $AK,CK,BK$ meet $k_{1},k_{2},k_{3}$ again at $D,E,F$ , respectively. If the areas of triangles $ABC,DBC,ECA,FAB$ are $u,x,y,z$ , respectively, prove that $$u^{2}=x^{2}+y^{2}+z^{2}.$$