ILL 1971 Problem 29

A rhombus with its incircle is given. At each vertex of the rhombus a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii $r_1,r_2$ , while the incircle has radius $r$ . Given that $r_1$ and $r_2$ are natural numbers and that $r_1r_2=r$ , find $r_1,r_2,$ and $r$ .