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Vector geometry with excenter

Source: Romanian District Olympiad 2023 9.1

March 11, 2023

geometryvector geometry

Problem Statement

Consider the triangle

ABC{}

and let

I_A{}

be its

A{}

-excenter. Let

M,N

and

P{}

be the projections of

I_A{}

onto the lines

AC,BC{}

and

AB{}

respectively. Prove that if

\overrightarrow{I_AM}+\overrightarrow{I_AP}=\overrightarrow{I_AN}

then

ABC{}

is an equilateral triangle.

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