MathDB

Circle and 3-4-5

Source: 2014 AIME I Problem 15

March 14, 2014

trigonometryanalytic geometrygeometrycyclic quadrilateralPythagorean TheoremAIME

Problem Statement

In

\triangle ABC

,

AB = 3

,

BC = 4

, and

CA = 5

. Circle

\omega

intersects

\overline{AB}

at

E

and

B

,

\overline{BC}

at

B

and

D

, and

\overline{AC}

at

F

and

G

. Given that

EF=DF

and

\tfrac{DG}{EG} = \tfrac{3}{4}

, length

DE=\tfrac{a\sqrt{b}}{c}

, where

a

and

c

are relatively prime positive integers, and

b

is a positive integer not divisible by the square of any prime. Find

a+b+c

.

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