BG-MO 1965 Problem 3

In the triangle $ABC$ , angle bisector $CD$ intersects the circumcircle of $ABC$ at the point $K$ .

(a) Prove the equalities: $$\frac1{ID}-\frac1{IK}=\frac1{CI},\enspace\frac{CI}{ID}-\frac{ID}{DK}=1$$ where $I$ is the center of the inscribed circle of triangle $ABC$ . (b) On the segment $CK$ some point $P$ is chosen whose projections on $AC,BC,AB$ respectively are $P_1,P_2,P_3$ . The lines $PP_3$ and $P_1P_2$ intersect at a point $M$ . Find the locus of $M$ when $P$ moves around segment $CK$ .