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Prove that the lines $BP$ and $CQ$ are perpendicular.

Source: Moldova TST 2022

April 1, 2022

geometry

Problem Statement

Let

A

be a point outside of the circle

\Omega

. Tangents from

A

touch

\Omega

in points

B

and

C

. Point

C

, collinear with

A

and

P

, is between

A

and

P

, such that the circumcircle of triangle

ABP

intersects

\Omega

again in point

E

. Point

Q

is on the segment

BP

, such that

\angle PEQ=2 \cdot \angle APB

. Prove that the lines

BP

and

CQ

are perpendicular.

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