PAMO 2006 Problem 6

Let $ABC$ be a right angled triangle at $A$ . Denote $D$ the foot of the altitude through $A$ and $O_1, O_2$ the incentres of triangles $ADB$ and $ADC$ . The circle with centre $A$ and radius $AD$ cuts $AB$ in $K$ and $AC$ in $L$ . Show that $O_1, O_2, K$ and $L$ are on a line.