49th Mathematical Olympiad in Ukraine

A parallelogram $ABCD$ with $AC>BD$ is given. Let $k_1$ be the circle with diameter $AC$ and $k_2$ the circle with diameter $DC$. $k_1$ meets line $AB$ at the point $E$, $k_2$ meets the line $AC$ at the points $C$ and $O$, and line $AD$ at the point $F$. Suppose that $AO=a$, $FO=b$ and $\angle BAC=45^{\circ }$. Find the ratio of the areas of the triangles $AOE$ and $COF$.

Answer: ${\left(\frac{a}{b}\right)}^2$.