ELMO SL 2013 G3

In $\triangle ABC$ , a point $D$ lies on line $BC$ . The circumcircle of $ABD$ meets $AC$ at $F$ (other than $A$ ), and the circumcircle of $ADC$ meets $AB$ at $E$ (other than $A$ ). Prove that as $D$ varies, the circumcircle of $AEF$ always passes through a fixed point other than $A$ , and that this point lies on the median from $A$ to $BC$ .

Proposed by Allen Liu