Question 2
Let $ABC$ be a right triangle with $\angle B=90^\circ$. A point $D$ lies on line $CB$ with $B$ between $D,C$. Let $E$ be the midpoint of $AD$, and let $F$ be the second intersection of the circumcircles of $ACD$ and $BDE$. Prove that, as $D$ varies, line $EF$ passes through a fixed point.
Editorial clarification: $F$ is a distinct second intersection, so exclude $DB=BC$, when the two circles are tangent at $D$. The fixed-point assertion applies to every well-defined position. The source wording is retained above; the limiting line at the excluded position is described in the solution.
Editorial clarification: $F$ is a distinct second intersection, so exclude $DB=BC$, when the two circles are tangent at $D$. The fixed-point assertion applies to every well-defined position. The source wording is retained above; the limiting line at the excluded position is described in the solution.