MathDB

Let

ABCD

be a parallelogram with

\angle BAD < 90^{\circ}

. A circle tangent to sides

\overline{DA}

\overline{AB}

, and

\overline{BC}

intersects diagonal

\overline{AC}

at points

P

and

Q

with

AP < AQ

, as shown. Suppose that

AP = 3

PQ = 9

, and

QC = 16

. Then the area of

ABCD

can be expressed in the form

m\sqrt n

, where

m

and

n

are positive integers, and

n

is not divisible by the square of any prime. Find

m+n

.
[asy] defaultpen(linewidth(0.6)+fontsize(11)); size(8cm); pair A,B,C,D,P,Q; A=(0,0); label("

A

", A, SW); B=(6,15); label("

B

", B, NW); C=(30,15); label("

C

", C, NE); D=(24,0); label("

D

", D, SE); P=(5.2,2.6); label("

P

", (5.8,2.6), N); Q=(18.3,9.1); label("

Q

", (18.1,9.7), W); draw(A--B--C--D--cycle); draw(C--A); draw(Circle((10.95,7.45), 7.45)); dot(A^^B^^C^^D^^P^^Q); [/asy]

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