# What Is This Point?

Geometry Level 5

Let $$\triangle ABC$$ be an acute angled triangle with side lengths $$AB= 5, BC= 7, CA= 8.$$ Let $$D$$ be the foot of perpendicular from $$A$$ to $$BC,$$ and let $$O$$ be its circumcenter. The feet of perpendiculars from $$O$$ to $$AB$$ and $$AC$$ intersect $$AD$$ at points $$Q$$ and $$P$$ respectively. Let $$S$$ be the circumcenter of $$\triangle OPQ.$$ If $$\cos (\angle CAS) = a \sqrt{\dfrac{b}{c}}$$ for some squarefree coprime positive integers $$b,c$$ and a positive integer $$a,$$ find $$a+b+c.$$

Details and assumptions
- This problem is adapted from an ARO 10th grade geometry problem.

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