# 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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