Problem #52

Given square ABCD, draw segments AM, BN, CK,
DL where K, L, M, and N are the
midpoints of segments AB, BC, CD, and DA
respectively as shown in the figure below.
Find the ratio of the area of the shaded square to the area of square
ABCD

What is the ratio if K, L, M, and N are chosen
so that AK/KB = BL/LC = CM/MD =
DN /NA = x? In the original problem, x = 1 and
in the figure below, it's 2.