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equal lengths ,tangents and angle bisectors

Source: Nigerian Mathematics olympiad round 2 2021 problem 5

February 15, 2021

geometrycyclic quadrilateralcircumcircleangle bisector

Problem Statement

let

ABCD

be a cyclic quadrilateral with

E

,an interior point such that

AB=AD=AE=BC

. Let

DE

meet the circumcircle of

BEC

again at

F

. Suppose a common tangent to the circumcircle of

BEC

and

DEC

touch the circles at

F

and

G

respectively. Show that

GE

is the external angle bisector of angle

BEF