# A lot of tangency points

Geometry Level 4

The triangle $$ABC$$ is inscribed in the circle $$\mathcal{C}$$. Circle $$\mathcal{C}_1$$ is tangent to the minor arc $$AB$$ of $$\mathcal{C}$$ and to side $$AC$$ at point $$P$$. Circle $$\mathcal{C}_2$$ is tangent to the minor arc $$CB$$ of $$\mathcal{C}$$ and to side $$AC$$ at point $$Q$$. From point $$B$$ draw the tangents $$BR$$ and $$BS$$ to circles $$\mathcal{C}_1$$ and $$\mathcal{C}_2$$, respectively. It is known that $$BA=22$$, $$BC=23$$, $$BR=16$$, $$BS=17$$, $$PQ=15$$. $$AC$$ can be written as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a+b$$?

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