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Isosceles trapezoid, parralel lines and a tangency

Source: St Petersburg 2021 9.5

December 23, 2021

geometry

Problem Statement

Given is an isosceles trapezoid

ABCD

, such that

AD

and

BC

are bases and

AD=2AB

, and it is inscribed in a circle

c

. Points

E

and

F

are selected on a circle

c

so that

AC

||

DE

and

BD

||

AF

. The line

BE

intersects lines

AC

and

AF

at points

X

and

Y

, respectively. Prove that the circumcircles of triangles

BCX

and

EFY

are tangent to each other.

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