# Classical Triangle Centers

Geometry Level 5

In a isosceles triangle $$ABC$$ with $$\angle B = \angle C$$, the incenter $$I$$ is in the midpoint of the line segment determined by the orthocenter $$H$$ and the barycenter $$G$$. The length of line segment $$\overline{AG}$$ is less than the length of line segment $$\overline{AH}$$. The lengths of the line segments $$\overline{HI}$$ and $$\overline{IG}$$ are $$d$$.

If the area of the $$ABC$$ triangle can be expressed as

$\frac{ad^2\sqrt{b}}{c}$

with $$a,b,c$$ being positive integers with $$a,c$$ coprime and $$b$$ square-free, find $$a+b+c$$.

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