# Find The Distance Between Their Centers

Geometry Level 5

In $$\triangle ABC,$$ $$BC= 5, CA= 7, AB= 8.$$ Let $$\omega$$ and $$\Gamma$$ denote the circumcircle and incircle of $$\triangle ABC$$ respectively. A circle $$\delta$$ centered at point $$P$$ is externally tangent to $$\Gamma$$ and internally tangent to $$\omega$$ at $$A.$$ Another circle centered at $$Q$$ is internally tangent to both $$\omega$$ and $$\delta$$ at $$A.$$ The length of $$PQ$$ can be expressed as $$\dfrac{a}{b\sqrt{c}}$$ for some coprime positive integers $$a,b$$ and a prime $$c.$$ Find $$a+b+c.$$

Details and assumptions

• This problem is inspired by an old USAMO problem.

• This diagram is not mine. I took it off from the AoPS thread.

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