find the area of the octagon

Question

Find the area of the colored octagon (tell the ratio between square $ABCD$ and the colored polygon).

Square $ABCD$ is a perfect square, and $E,F,G,H$ are the midpoints of the line they are at.

(Hint: The top-left dot of the octagon is the center of gravity of triangle $ABD$.)

This question is derived from IMTS R19 Question 5.

Answer 1

Hint: referring to the figure below and letting $BC=4$, can you show $NE=1$, $XN=1$ and calculate the area of triangle $XMN$?

$\hspace{5cm}$

Can you generalize it for any square with $BC=a$? You should get a formula for the areas in terms of $a$.