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symmetric lines with respect to angle bisector

Source: Azerbaijan EGMO TST 2021, D1 P2

August 11, 2023

geometryisogonal linesAZE EGMO TST

Problem Statement

Let

\omega

be a circle with center

O,

and let

A

be a point with tangents

l_2

and

AQ

to the circle. Denote by

K

the intersection point of

AO

and

PQ.

l_1

and

l_2

are two lines passing through

A

that intersect

\omega.

Call

B

the intersection point of

l_1

with

\omega,

which is located nearer to

A

on

l_1.

Call

C

the intersection point of

l_2

with

\omega,

which is located further to

A

on

l_2.

Prove that

\angle PAB = \angle QAC

if and only if the points

B, K, C

are on line.

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