MathDB

Circumcircle Intersections

Source: 2019 AIME I #13

March 14, 2019

AIMEAIME Igeometrycircumcircle2019 AIME Itrigonometrypower of a point

Problem Statement

Triangle

ABC

has side lengths

AB=4

,

BC=5

, and

CA=6

. Points

D

and

E

are on ray

AB

with

DF=2

. The point

F \neq C

is a point of intersection of the circumcircles of

\triangle ACD

and

\triangle EBC

satisfying

DF=2

and

EF=7

. Then

BE

can be expressed as

\tfrac{a+b\sqrt{c}}{d}

, where

a

,

b

,

c

, and

d

are positive integers such that

a

and

d

are relatively prime, and

c

is not divisible by the square of any prime. Find

a+b+c+d

.

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