Let $ABC$ be a triangle and let $A' \in BC$, $B' \in CA$ and $C' \in AB$ be three collinear points. a) Prove that each pair of circles of diameters $AA'$, $BB'$ and $CC'$ has the same radical axis; b) Prove that the circumcenter of the triangle formed by the intersections of the lines $AA' , BB'$ and $CC'$ lies on the common radical axis found above.