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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