# 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?

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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 $a,b,c$.

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You approached from nice point, Really awesome.Thanks – Mathlover Apr 17 '12 at 23:10
@Mathlover: You are welcome. – Aryabhata Apr 17 '12 at 23:14