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Geometry

Source: 8th European Mathematical Cup 2019 Junior Q3

December 26, 2019

geometry

Problem Statement

Let

ABC

be a triangle with circumcircle

\omega

. Let

l_B

and

l_C

be two lines through the points

B

and

C

, respectively, such that

l_B \parallel l_C

. The second intersections of

l_B

and

l_C

with

\omega

are

D

and

E

, respectively. Assume that

D

and

E

are on the same side of

BC

as

A

. Let

DA

intersect

l_C

at

F

and let

EA

intersect

l_B

at

G

. If

O

,

O_1

and

O_2

are circumcenters of the triangles

ABC

,

ADG

and

AEF

, respectively, and

P

is the circumcenter of the triangle

OO_1O_2

, prove that

l_B \parallel OP \parallel l_C

.
Proposed by Stefan Lozanovski, Macedonia

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