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Steiner's Theorem Returns

Source: 2012 AIME I Problem 12

March 16, 2012

trigonometryratioLaTeXAMCAIMEnumber theoryrelatively prime

Problem Statement

Let

\triangle ABC

be a right triangle with right angle at

C

. Let

D

and

E

be points on

\overline{AB}

with

D

between

A

and

E

such that

\overline{CD}

and

\overline{CE}

trisect

\angle C

. If

\frac{DE}{BE} = \frac{8}{15}

, then

\tan B

can be written as

\frac{m\sqrt{p}}{n}

, where

m

and

n

are relatively prime positive integers, and

p

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

m+n+p

.

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