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Pentagon geometry

Source: Russian TST 2019, Day 8 P1 (Groups A & B)

March 27, 2023

geometrypentagon

Problem Statement

A convex pentagon

APBCQ

is given such that

AB < AC

. The circle

\omega

centered at point

A{}

passes through

P{}

and

Q{}

and touches the segment

BC

at point

R{}

. Let the circle

\Gamma

centered at the point

O{}

be the circumcircle of the triangle

ABC

. It is known that

AO \perp P Q

and

\angle BQR = \angle CP R

. Prove that the tangents at points

P{}

and

Q{}

to the circle

\omega

intersect on

\Gamma

.

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