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Three tangents lemma plus a tangent circle

Source: 2021 Macedonian Team Selection Test P6

May 30, 2021

geometry

Problem Statement

Let

ABC

be an acute triangle such that

AB<AC

with orthocenter

H

. The altitudes

BH

and

CH

intersect

AC

and

AB

at

B_{1}

and

C_{1}

. Denote by

M

the midpoint of

BC

. Let

l

be the line parallel to

BC

passing through

A

. The circle around

CMC_{1}

meets the line

l

at points

X

and

Y

, such that

X

is on the same side of the line

AH

as

B

and

Y

is on the same side of

AH

as

C

. The lines

MX

and

MY

intersect

CC_{1}

at

U

and

V

respectively. Show that the circumcircles of

MUV

and

B_{1}C_{1}H

are tangent.
Proposed by Nikola Velov

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