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equal angles wanted, two intersecting circles both tangent to given rays

Source: Ukrainian Geometry Olympiad 2020, XI p3

April 27, 2020

geometryequal anglescirclesTangents

Problem Statement

The angle

POQ

is given (

OP

and

OQ

are rays). Let

M

and

N

be points inside the angle

POQ

such that

\angle POM = \angle QON

and

\angle POM < \angle PON

. Consider two circles: one touches the rays

OP

and

ON

, the other touches the rays

OM

and

OQ

. Denote by

B

and

C

the points of their intersection. Prove that

\angle POC = \angle QOB