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junior perpenicularity, 2 circles related

Source: Greece Junior Math Olympiad 2024 p2

March 2, 2024

geometryperpendicular

Problem Statement

Consider an acute triangle

ABC

and it's circumcircle

\omega

. With center

A

, we construct a circle

\gamma

that intersects arc

AB

of circle

\omega

, that doesn't contain

C

, at point

D

and arc

AC

, that doesn't contain

B

, at point

E

. Suppose that the intersection point

K

of lines

BE

and

CD

lies on circle

\gamma

. Prove that line

AK

is perpendicular on line

BC

.

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