How to find x that defined in the picture? 
$O$ is center of the circle that surrounds the ABC triangle.
$|EF| // |BC|$ 
we only know $a,b,c$ 
$(a=|BC|, b=|AC|,c=|AB|)$
$x=|EG|=?$
Could you please give me hand to see an easy way to find the x that depends on given a,b,c? 

This can be done using trigonometry.
Let $D$ be the foot of perpendicular from $O$ to $BC$.
Then we have that $\angle{BOD} = \angle{BAC} (= \alpha, \text{say})$.
Let $\angle{CBA} = \beta$.
Let the radius of the circumcircle be $R$.
Let $I$ be the foot of perpendicular from $G$ on $BC$.
Then we have that $DB = R\sin \alpha$, by considering $\triangle BOD$
$GI = OD = R \cos \alpha$.
By considering $\triangle BGI$,  $BI = GI \cot \beta = R \cos \alpha \cot \beta$.
Thus $x = R - OG = R - (BD - BI) = R - R\sin \alpha + R \cos \alpha \cot \beta$.
Now, $R$ and trigonometric functions of $\alpha$ and $\beta$ can be expressed in terms of $a,b,c$.
