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PAMO Problem 2: Equality of angles in a geometry problem with tangents

Source: 2021 Pan-African Mathematics Olympiad, Problem 2

May 24, 2021

geometrycircumcirclegeometric transformationreflectionPAMO

Problem Statement

Let

\Gamma

be a circle,

P

be a point outside it, and

A

and

B

the intersection points between

\Gamma

and the tangents from

P

to

\Gamma

. Let

K

be a point on the line

AB

, distinct from

A

and

\angle PBT=\angle P'KA

and let

T

be the second intersection point of

\Gamma

and the circumcircle of the triangle

PBK

.Also, let

P'

be the reflection of

P

in point

A

. Show that

\angle PBT=\angle P'KA

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