Circumcircle intersection ratio on lines

Collinear points $A$, $B$, and $C$ are given in the Cartesian plane such that $A=(a, 0)$ lies along the $x$-axis, $B$ lies along the line $y=x$, $C$ lies along the line $y=2x$, and $AB / BC = 2$. If $D=(a, a)$, the circumcircle of triangle $ADC$ intersects $y=x$ again at $E$, and ray $AE$ intersects $y=2x$ at $F$, evaluate $AE / EF$.