Expected perimeter of a triangle inscribed at random

Pick three points $A$, $B$, $C$ independently and uniformly on the unit circle. The expected perimeter of triangle $ABC$ can be written as $\tfrac{a}{\pi}$ with $a$ rational. What is $a$?

$a=12$. The three sides have the same distribution, so the expected perimeter is three times the expected chord length. With central angle $\Theta\sim\text{Unif}(0,2\pi)$ the chord is $2\sin(\Theta/2)$, whose mean is $\tfrac{4}{\pi}$, so the perimeter averages $\tfrac{12}{\pi}$.