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Circles, cords, and symmetry

Source: Poland Math Olympiad 1993 Round 1 #4

June 3, 2023

symmetry

Problem Statement

Given is a circle with center

l

, point

A

inside the circle and a chord

OA

which is not a diameter and passing through

A

. The lines

p

and

q

are tangent to the given circle at

P

and

Q

respectively. The line

l

passing through

A

and perpendicular to

OA

intersects the lines

p

and

q

at

K

and

L

respectively. Prove that

|AK| = |AL|

.

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