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Circumcenter midpoint of segment

Source: Cono Sur Olympiad 2017, problem 4

August 21, 2017

geometrycono surcircumcircle

Problem Statement

Let

ABC

an acute triangle with circumcenter

O

. Points

X

and

Y

are chosen such that:
[*]

\angle XAB = \angle YCB = 90^\circ

[/*] [*]

\angle ABC = \angle BXA = \angle BYC

[/*] [*]

X

and

C

are in different half-planes with respect to

AB

[/*] [*]

Y

and

A

are in different half-planes with respect to

BC

[/*]
Prove that

O

is the midpoint of

XY

.

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