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Cyclic pentagon

Source: 2023 Kürschák Mathematics Competition/3

October 7, 2023

geometrytangent circlescircumcircle

Problem Statement

Given is a convex cyclic pentagon

AD

and a point

P

inside it, such that

AB=AE=AP

and

BC=CE

. The lines

AD

and

BE

intersect in

Q

. Points

BEP

and

S

are on segments

CP

and

BP

such that

DR=QR

and

SR||BC

. Show that the circumcircles of

BEP

and

PQS

are tangent to each other.

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