Passel Of Techniques

Geometry Level 4

$ABC$ is a right triangle with $\angle B= 90^\circ$ .
$E$ and $F$ are the mid-points of $AB$ and $AC$ , respectively.
The incenter $I$ of triangle $ABC$ lies on the circumcircle of triangle $AEF$ ,

Find the ratio $\dfrac{BC}{AB}$ .

If this ratio can be expressed as $\dfrac pq$ , where $p$ and $q$ are coprime positive integers, submit your answer as $p+q$ .