Question 4
Let $ABCD$ be a square with centre $O$. Let $M$ lie on $AB$, distinct from $A,B$. The line through $M$ parallel to $BD$ meets $AC$ at $N$. Lines $DN$ and $CM$ meet at $E$, and $DN$ meets $AB$ at $F$.
a) Prove that angle $DEC$ is constant as $M$ varies.
b) Prove that $E$ lies on the circle with centre $O$ and radius $OA$.
c) Find the position of $M$ that maximizes $MF$.
a) Prove that angle $DEC$ is constant as $M$ varies.
b) Prove that $E$ lies on the circle with centre $O$ and radius $OA$.
c) Find the position of $M$ that maximizes $MF$.