MathDB

Geometry yet again

Source: 2006 AIME II 12

March 28, 2006

geometrytrigonometryAMCAIMEparallelogramgeometric transformationhomothety

Problem Statement

Equilateral

\triangle ABC

is inscribed in a circle of radius 2. Extend

\overline{AB}

through

B

to point

D

so that

AD=13

, and extend

\overline{AC}

through

C

to point

E

so that

AE=11

. Through

D

, draw a line

l_1

parallel to

\overline{AE}

, and through

E

, draw a line

{l}_2

parallel to

\overline{AD}

. Let

F

be the intersection of

{l}_1

and

{l}_2

. Let

G

be the point on the circle that is collinear with

A

and

F

and distinct from

A

. Given that the area of

\triangle CBG

can be expressed in the form

\frac{p\sqrt{q}}{r}

, where

p

,

q

, and

r

are positive integers,

p

and

r

are relatively prime, and

q

is not divisible by the square of any prime, find

p+q+r

.

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