Is it long?

Geometry Level 5

Consider \(\triangle ABC\) having sides \(AB=AC=14 \text{ cm}\) and \(BC=20 \text{ cm}\).

Let \(I\) be the incentre of \(\triangle ABC\) and \(G\) be the centroid of the triangle formed by joining the points at which the sides \(AB,BC\) and \(CA\) are tangent to the incircle of \(\triangle ABC\).

If \(IG=\dfrac{a\sqrt{b}}{c}\text{ cm}\) where \(\gcd (a,c)=1\) and \(b\) is square-free, then submit your answer as \(a+b+c\).

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