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Regional Olympiad - FBH 2015 Grade 12 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2015

September 23, 2018

geometrycircumcircleincenter

Problem Statement

Let

O

and

I

be circumcenter and incenter of triangle

ABC

. Let incircle of

ABC

touches sides

BC

CA

and

AB

in points

D

E

and

F

, respectively. Lines

FD

and

CA

intersect in point

P

, and lines

DE

and

AB

intersect in point

Q

. Furthermore, let

M

and

N

be midpoints of

PE

and

QF

. Prove that

OI \perp MN