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exists a circle tangent to three others

Source: 2020 MEMO I-3

August 30, 2020

geometrycircumcircleincentermemoMEMO 2020

Problem Statement

Let

ABC

be an acute scalene triangle with circumcircle

\omega

and incenter

I

. Suppose the orthocenter

H

of

BIC

lies inside

\omega

. Let

M

be the midpoint of the longer arc

BC

of

\omega

. Let

N

be the midpoint of the shorter arc

AM

of

\omega

. Prove that there exists a circle tangent to

\omega

at

N

and tangent to the circumcircles of

BHI

and

CHI

.

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