# Find The Length

Geometry Level 5

Let $$\triangle ABC$$ be a triangle with side lengths $$BC = 9, CA= 8, AB= 6.$$ Let $$O$$ and $$I$$ be the circumcenter and incenter of $$\triangle ABC$$ respectively. The incircle of $$\triangle ABC$$ touches $$BC$$ at point $$D.$$ Let $$X$$ be the second point of intersection of line $$AO$$ and the circumcircle of $$\triangle AID$$ (apart from $$A$$). Given that $$AX = \sqrt{\dfrac{a}{b}}$$ for some coprime positive integers $$a,b,$$ find $$a+b.$$

Details and assumptions

• The image shown is not accurate.

• This problem is adapted from a Russia 10th grade geometry problem.

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