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Circumcenter is a reflection of another circumcenter

Source: Croatian Mathematical Olympiad 2018

May 6, 2019

geometrytangentCircumcenterReflectionscircumcircle

Problem Statement

Let

k

be a circle centered at

O

. Let

\overline{AB}

be a chord of that circle and

M

its midpoint. Tangent on

k

at points

A

and

B

intersect at

T

. The line

\ell

goes through

T

, intersect the shorter arc

AB

at the point

C

and the longer arc

AB

at the point

D

, so that

|BC| = |BM|

. Prove that the circumcenter of the triangle

ADM

is the reflection of

O

across the line

AD

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