MathDB

IMOR 2017 - Problem 3

Source: 1st International Mathematical Olympic Revenge

July 22, 2017

IMORgeometry

Problem Statement

Let

ABC

be a triangle, and let

P

be a distinct point on the plane. Moreover, let

A'B'C'

be a homothety of

ABC

with ratio

2

and center

P

, and let

O

and

O'

be the circumcenters of

ABC

and

A'B'C'

, respectively. The circumcircles of

AB'C'

,

A'BC'

, and

A'B'C

meet at points

X

,

Y

, and

Z

, different from

A'

,

B'

, and

C'

. In a similar way, the circumcircles of

A'BC

,

AB'C

, and

ABC'

meet at

X'

,

Y'

, and

Z'

, different from

A

,

B

,

C

. Let

W

and

W'

be the circumcenters of

XYZ

and

X'Y'Z'

, respectively. Prove that

OW

is parallel to

O'W'

.
Proposed by Mateus Thimóteo, Brazil.

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