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Nice geometry from EMC

Source: 10th European Mathematical Cup - Problem S2

December 22, 2021

geometry

Problem Statement

Let

ABC

be a triangle and let

D, E

and

F

be the midpoints of sides

BC, CA

and

AB

, respectively. Let

X\ne A

be the intersection of

AD

with the circumcircle of

\Omega

. Let

\Omega

be the circle through

D

and

X

, tangent to the circumcircle of

ABC

. Let

Y

and

ZF

be the intersections of the tangent to

\Omega

at

D

with the perpendicular bisectors of segments

DE

and

DF

, respectively. Let

ABC

be the intersection of

YE

and

ZF

and let

G

be the centroid of

ABC

. Show that the tangents at

B

and

C

to the circumcircle of

ABC

and the line

PG

are concurrent.

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