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Sharky-devil point with cicumcenter lying on BC

Source: Azerbaijan IZhO TST 2021, P4

August 10, 2023

geometrycircumcirclesharky-deviltangency

Problem Statement

Let

ABC

be a triangle with incircle touching

BC, CA, AB

at

D, E, F,

respectively. Let

O

and

M

be its circumcenter and midpoint of

BC.

Suppose that circumcircles of

AEF

and

ABC

intersect at

X

for the second time. Assume

Y \neq X

is on the circumcircle of

ABC

such that

OMXY

is cyclic. Prove that circumcenter of

DXY

lies on

BC.

Proposed by tenplusten.

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