BG-MO 1991 Problem 1

Let $M$ be a point on the altitude $CD$ of an acute-angled triangle $ABC$ , and $K$ and $L$ the orthogonal projections of $M$ on $AC$ and $BC$ . Suppose that the incenter and circumcenter of the triangle lie on the segment $KL$ .

(a) Prove that $CD=R+r$ , where $R$ and $r$ are the circumradius and inradius, respectively. (b) Find the minimum value of the ratio $CM:CD$ .