Three small circles

Geometry Level 5

In triangle \(\triangle ABC\), let \(AB=c\), \(AC=b\) and \(BC=a\). Now, three circles with centers \(D\), \(E\) and \(F\) are drawn, such that all of them are tangent to the incircle of \(\triangle ABC\). The first one is also tangent to \(AB\) and \(AC\), the second one with \(AB\) and \(BC\), and the third one with \(AC\) and \(BC\).

The radii of these circles are \(\dfrac{73-\sqrt{145}}{36}\), \(\dfrac{66-8\sqrt{29}}{25}\) and \(18-8\sqrt{5}\), respectively.

If we know that all the sides of the triangle \(\triangle ABC\) are integers and \(a<b<c\), find \(10000a+100b+c\).

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