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Macedonia National Olympiad 2017 Problem 4

Source: Macedonia National Olympiad 2017

April 8, 2017

geometrycircumcircleangle bisectorincenter

Problem Statement

Let

O

be the circumcenter of the acute triangle

ABC

(

AB < AC

). Let

A_1

and

P

be the feet of the perpendicular lines drawn from

A

and

O

to

BC

, respectively. The lines

BO

and

CO

intersect

AA_1

in

D

and

E

, respectively. Let

F

be the second intersection point of

\odot ABD

and

\odot ACE

. Prove that the angle bisector od

\angle FAP

passes through the incenter of

\triangle ABC

.

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