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Another geometry with equilateral triangle

Source: 2006 AIME II 6

March 28, 2006

geometrytrigonometryAMCAIMEUSA(J)MOUSAMOsimilar triangles

Problem Statement

Square

ABCD

has sides of length 1. Points

E

and

F

are on

\overline{BC}

and

\overline{CD}

, respectively, so that

\triangle AEF

is equilateral. A square with vertex

B

has sides that are parallel to those of

ABCD

and a vertex on

\overline{AE}

. The length of a side of this smaller square is

\displaystyle \frac{a-\sqrt{b}}{c}

, where

a

,

b

, and

c

are positive integers and

b

is not divisible by the square of any prime. Find

a+b+c

.

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