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concentric circumcircles wanted, starting with isosceles trapezoid

Source: Iranian Geometry Olympiad 2021 IGO Elementary p4

January 25, 2022

geometryreflectiontrapezoidconcentric circles

Problem Statement

In isosceles trapezoid

ABCD

(

AB \parallel CD

) points

E

and

F

lie on the segment

CD

in such a way that

D, E, F

and

C

are in that order and

DE = CF

. Let

X

and

Y

be the reflection of

E

and

C

with respect to

AD

and

AF

. Prove that circumcircles of triangles

ADF

and

BXY

are concentric.
Proposed by Iman Maghsoudi - Iran