MathDB

In the diagram below,

ABCD

is a square. Point

E

is the midpoint of

\overline{AD}

. Points

F

and

G

lie on

\overline{CE}

, and

H

and

J

lie on

\overline{AB}

and

\overline{BC}

, respectively, so that

FGHJ

is a square. Points

K

and

L

lie on

\overline{GH}

, and

M

and

N

lie on

\overline{AD}

and

\overline{AB}

, respectively, so that

KLMN

is a square. The area of

KLMN

is 99. Find the area of

FGHJ

. [asy] pair A,B,C,D,E,F,G,H,J,K,L,M,N; B=(0,0); real m=7*sqrt(55)/5; J=(m,0); C=(7*m/2,0); A=(0,7*m/2); D=(7*m/2,7*m/2); E=(A+D)/2; H=(0,2m); N=(0,2m+3*sqrt(55)/2); G=foot(H,E,C); F=foot(J,E,C); draw(A--B--C--D--cycle); draw(C--E); draw(G--H--J--F); pair X=foot(N,E,C); M=extension(N,X,A,D); K=foot(N,H,G); L=foot(M,H,G); draw(K--N--M--L); label("

A

",A,NW); label("

B

",B,SW); label("

C

",C,SE); label("

D

",D,NE); label("

E

",E,dir(90)); label("

F

",F,NE); label("

G

",G,NE); label("

H

",H,W); label("

J

",J,S); label("

K

",K,SE); label("

L

",L,SE); label("

M

",M,dir(90)); label("

N

",N,dir(180)); [/asy]

Problem Statement