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Geometry with line passing through circumcenter

Source: Russian TST 2014, Day 8 P3 (Groups A & B)

January 8, 2024

geometryangles

Problem Statement

On the sides

AB{}

and

AC{}

of the acute-angled triangle

ABC{}

the points

M{}

and

N{}

are chosen such that

MN

passes through the circumcenter of

ABC.

Let

P{}

and

Q{}

be the midpoints of the segments

CM{}

and

BN{}.

Prove that

\angle POQ=\angle BAC.

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