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Albania BMO TST Problem 4

Source:

April 1, 2017

geometry

Problem Statement

The incircle of

\triangle A_{0}B_{0}C_{0}

, meets legs

B_{0}C_{0}

,

C_{0}A_{0}

,

A_{0}B_{0}

, respectively on points

A

,

B

,

C

, and the incircle of

\triangle ABC

, with center

I

, meets legs

BC

,

CA

,

AB

, on points

A_{1}

,

B_{1}

,

C_{1}

, respectively. We write with

\sigma (ABC)

, and

\sigma (A_{1}B_{1}C_{1})

the areas of

\triangle ABC

, and

\triangle A_{1}B_{1}C_{1}

respectively. Prove that if

\sigma (ABC)=2 \sigma (A_{1}B_{1}C_{1})

, then lines

AA_{0}

,

BB_{0}

,

CC_{0}

are concurrent.

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