ELMO SL 2013 G11

Let $\triangle ABC$ be a nondegenerate isosceles triangle with $AB=AC$ , and let $D, E, F$ be the midpoints of $BC, CA, AB$ respectively. $BE$ intersects the circumcircle of $\triangle ABC$ again at $G$ , and $H$ is the midpoint of minor arc $BC$ . $CF\cap DG=I, BI\cap AC=J$ . Prove that $\angle BJH=\angle ADG$ if and only if $\angle BID=\angle GBC$ .

Proposed by David Stoner