# The more the merrier ....

Geometry Level 5

Suppose we have $$4$$ circles $$P, Q, R, S$$ each of radius $$1$$ such that $$Q$$ is centered at $$(0,1)$$, $$R$$ at $$(2,1)$$, $$S$$ at $$(4,1)$$ and with $$P$$ lying in the first quadrant tangent to both $$R$$ and $$S$$.

Form a triangle $$\Delta ABC$$ that circumscribes the $$4$$ circles such that $$AB$$ is tangent to $$P$$ and $$Q$$, $$AC$$ is tangent to $$P$$ and $$S$$, and $$BC$$ is tangent to $$Q, R$$ and $$S$$.

The perimeter of $$\Delta ABC$$ can be written as $$a + b\sqrt{c}$$, where $$a,b,c$$ are positive integers with $$c$$ being square-free. Find $$a + b + c$$.

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