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tangent (ART) , (PEF) wanted

Source: 2023 IGO Adcanced P5

January 18, 2024

geometrytangent circles

Problem Statement

In triangle

BC

points

M

and

N

are the midpoints of sides

AC

and

AB

, respectively and

D

is the projection of

A

into

BC

. Point

DR

is the circumcenter of

ABC

and circumcircles of

E

,

DMN

intersect at points

R, T

. Lines

K

,

DR

intersect line

MN

at

E

and

F

, respectively. Lines

CT

,

BR

intersect at

K

. A point

P

lies on

KD

such that

PK

is the angle bisector of

\angle BPC

. Prove that the circumcircles of

ART

and

PEF

are tangent.
Proposed by Mehran Talaei - Iran

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