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AI perpendicular to PQ and AI=PQ

Source: INMO Problem 5

January 15, 2017

geometry

Problem Statement

Let

ABC

be a triangle with

\angle{A}=90^{\circ}

and

AB<AC

. Let

AD

be the altitude from

A

on to

BC

, Let

P,Q

and

I

denote respectively the incentres of triangle

ABD,ACD

and

ABC

. Prove that

AI

is perpendicular to

PQ

and

AI=PQ

.

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