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


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


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