SE-SKOLORNAS 2012 Problem 3

The catheti $AC$ and $BC$ in a right-angled triangle $ABC$ have lengths $b$ and $a$ , respectively. A circle centered at $C$ is tangent to hypotenuse $AB$ at point $D$ . The tangents to the circle through points $A$ and $B$ intersect the circle at points $E$ and $F$ , respectively (where $E$ and $F$ are both different from $D$ ). Express the length of the segment $EF$ in terms of $a$ and $b$ .