Question 3
(a) Let $ABCD$ be a convex quadrilateral, $I$ the midpoint of $AC$, and $M$ a point on $AC$ such that $AM=\sqrt2\,AI$. The line through $M$ parallel to $BC$ meets $AB$ at $E$, and the line through $M$ parallel to $DC$ meets $AD$ at $F$. Prove $S_{IEF}=\tfrac12S_{MDB}$. (b) In acute triangle $ABC$, $AM$ is a median ($M\in BC$), $BE$ is an altitude ($E\in AC$), and $K=AM\cap BE$. Given $AM=BE$, prove $AC=\sqrt3\,BK$.
