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# Geometry Problem

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In the diagram, four circles of radius 1 with centres $$P, Q, R,$$ and $$S$$ are tangent to one another and to the sides of $$\triangle ABC$$, as shown.

The radius of the circle with center $$R$$ is decreased so that

the circle with center $$R$$ remains tangent to $$BC$$,

the circle with center $$R$$ remains tangent to the other three circles, and

the circle with center $$P$$ becomes tangent to the other three circles.

The radii and tangencies of the other three circles stay the same. This changes the size and shape of $$\triangle ABC$$.  $$r$$ is the new radius of the circle with center $$R$$$$r$$ is of the form $$\frac{a+\sqrt{b}}{c}$$. Find $$a+b+c$$.

Jun 6, 2021