Problem:
In the diagram below, ABCD is a square. Point E is the midpoint of AD. Points F and G lie on CE, and H and J lie on AB and BC, respectively, so that FGHJ is a square. Points K and L lie on GH, and M and N lie on AD and AB, respectively, so that KLMN is a square. The area of KLMN is 99. Find the area of FGHJ.
Solution:
Let AE=s, so CD=2s, and CE=5s. Note that △CDE,△JFC,△HBJ, △NKH, and △MAN are similar to each other. Let x=FG and y=KL. Then 2s=BC=BJ+JC=5x+2x5, so x=51+252s=745s. Then AH=2s−HB=2s−52x=76s. Hence 76s=AH=AN+NH=5y+2y5, so y=51+2576s=49125s. The ratio of the areas of squares FGHJ and KLMN is (yx)2=(49125s745s)2=949. Thus if square KLMN has area 99, square FGHJ has area 99⋅949=539.
OR
With x and y defined as above, note that AN+NH+HB=BJ+JC so
51y+25y+52x=51x+25x
Thus
(51+25)y=(51+25−52)x
Then the ratio of the areas of squares FGHJ and KLMN is
