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parallel chords of pairs of intersecting circles

Source: 2015 Sharygin Geometry Olympiad Correspondence Round P21

August 2, 2018

geometrycirclescyclic quadrilateral

Problem Statement

A quadrilateral

ABCD

is inscribed into a circle

\omega

with center

O

. Let

M_1

and

M_2

be the midpoints of segments

AB

and

X_2

respectively. Let

\Omega

be the circumcircle of triangle

OM_1M_2

. Let

X_1

and

X_2

be the common points of

\omega

and

\Omega

and

Y_1

and

Y_2

the second common points of

\Omega

with the circumcircles of triangles

CDM_1

and

ABM_2

. Prove that

X_1X_2 // Y_1Y_2

.

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