IMAC 2008 Problem 2

In the $ABC$ triangle, the bisector of $A$ intersects the $[BC]$ at the point $A_ {1}$ , and the circle circumscribed to the triangle $ABC$ at the point $A_ {2}$ . Similarly are defined $B_ {1}$ and $B_ {2}$ , as well as $C_ {1}$ and $C_ {2}$ . Prove that $$\frac {A_{1}A_{2}}{BA_{2} + A_{2}C} + \frac {B_{1}B_{2}}{CB_{2} + B_{2}A} + \frac {C_{1}C_{2}}{AC_{2} + C_{2}B} \geq \frac {3}{4}$$