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O_1O_2 __|_ AB, starting with intersecting circles

Source: 2015 Latvia BW TST P7

December 16, 2022

perpendiculargeometry

Problem Statement

Two circle

\Gamma_1

and

\Gamma_2

intersect at points

A

and

B

, point

P

is not on the line

AB

. Line

AP

intersects again

\Gamma_1

and

\Gamma_2

at points

K

and

L

respectively, line

BP

intersects again

\Gamma_1

and

\Gamma_2

at points

M

and

N

respectively and all the points mentioned so far are different. The centers of the circles circumscribed around the triangles

KMP

and

LNP

are

O_1

and

O_2

respectively. Prove that

O_1O_2

is perpendicular to

AB

.

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