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Triangle geometry with a really weird condition on side lengths

Source: Germany 2022, Problem 3

June 25, 2022

geometrycircumcircleincircletouching circlessidelengthsgeometry proposed

Problem Statement

Let

M

and

N

be the midpoints of segments

BC

and

AC

of a triangle

ABC

, respectively. Let

Q

be a point on the line through

N

parallel to

BC

such that

Q

and

C

are on opposite sides of

AB

and

\vert QN\vert \cdot \vert BC\vert=\vert AB\vert \cdot \vert AC\vert

.
Suppose that the circumcircle of triangle

AQN

intersects the segment

MN

a second time in a point

T \ne N

. Prove that there is a circle through points

T

and

N

touching both the side

BC

and the incircle of triangle

ABC

.

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