# Are you in or are you out

Geometry Level 5

Let $ABCP$ be a quadrilateral inscribed in a circle $\Gamma$ such that $PB=PC$ and $\angle BAC$ is obtuse. Let $I$ be the incenter of triangle $ABC$ and suppose that line $PI$ intersects again $\Gamma$ at point $J$ ($J$ belongs to the major arc $BC$). If $BJ=10$, $JC=17$ and $\sin \angle BJC=\frac{77}{85}$, then the value of $\frac{AB}{AC}$ can be written as $\frac{a}{b}$, where $a$ and $b$ are coprime positive integers. What is the value of $a+b$?

Details and assumptions

The order of the vertices is $A, B, C, P$. In particular, $P$ lies on the minor arc of $BC$ which contains $A$.

×