2021 April MIMC 10 Problems/Problem 22

In the diagram, $ABCD$ is a square with area $6+4\sqrt{2}$ . $AC$ is a diagonal of square $ABCD$ . Square $IGED$ has area $11-6\sqrt{2}$ . Given that point $J$ bisects line segment $HE$ , and $AE$ is a line segment. Extend $EG$ to meet diagonal $AC$ and mark the intersection point $H$ . In addition, $K$ is drawn so that $JK//EC$ . $FH^2$ can be represented as $\frac{a+b\sqrt{c}}{{d}}$ where $a,b,c,d$ are not necessarily distinct integers. Given that $gcd(a,b,d)=1$ , and $c$ does not have a perfect square factor. Find $a+b+c+d$ .