ILL 1986 Problem 50

Let $D$ be the point on the side $BC$ of the triangle $ABC$ such that $AD$ is the bisector of $\angle CAB$ . Let $I$ be the incenter of $ABC.$ (a) Construct the points $P$ and $Q$ on the sides $AB$ and $AC$ , respectively, such that $PQ$ is parallel to $BC$ and the perimeter of the triangle $APQ$ is equal to $k \cdot BC$ , where $k$ is a given rational number.

*(b)*Let $R$ be the intersection point of $PQ$ and $AD$ . For what value of $k$ does the equality $AR = RI$ hold?

(c) In which case do the equalities $AR = RI = ID$ hold?