MathDB

On square

ABCD,

points

E,F,G,

and

H

lie on sides

\overline{AB},\overline{BC},\overline{CD},

and

\overline{DA},

respectively, so that

\overline{EG} \perp \overline{FH}

and

EG=FH = 34.

Segments

\overline{EG}

and

\overline{FH}

intersect at a point

P,

and the areas of the quadrilaterals

AEPH, BFPE, CGPF,

and

DHPG

are in the ratio

269:275:405:411.

Find the area of square

ABCD

.
[asy] size(200); defaultpen(linewidth(0.8)+fontsize(10.6)); pair A = (0,sqrt(850)); pair B = (0,0); pair C = (sqrt(850),0); pair D = (sqrt(850),sqrt(850)); draw(A--B--C--D--cycle); dotfactor = 3; dot("

A

",A,dir(135)); dot("

B

",B,dir(215)); dot("

C

",C,dir(305)); dot("

D

",D,dir(45)); pair H = ((2sqrt(850)-sqrt(120))/6,sqrt(850)); pair F = ((2sqrt(850)+sqrt(306)+7)/6,0); dot("

H

",H,dir(90)); dot("

F

",F,dir(270)); draw(H--F); pair E = (0,(sqrt(850)-6)/2); pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2); dot("

E

",E,dir(180)); dot("

G

",G,dir(0)); draw(E--G); pair P = extension(H,F,E,G); dot("

P

",P,dir(60)); label("

w

", (H+E)/2,fontsize(15)); label("

x

", (E+F)/2,fontsize(15)); label("

y

", (G+F)/2,fontsize(15)); label("

z

", (H+G)/2,fontsize(15)); label("

w:x:y:z=269:275:405:411

",(sqrt(850)/2,-4.5),fontsize(11)); [/asy]

Problem Statement