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prove a quadrilateral is cyclic

Source: Mathematics Regional Olympiad of Mexico Southeast 2018 P3

October 23, 2021

geometrycircumcircleparallel

Problem Statement

Let

ABC

a triangle with circumcircle

\Gamma

and

R

a point inside

ABC

such that

\angle ABR=\angle RBC

. Let

\Gamma_1

and

\Gamma_2

the circumcircles of triangles

ARB

and

CRB

respectly. The parallel to

E

that pass through

R

, intersect

\Gamma

in

D

and

E

, with

D

on the same side of

BR

that

A

and

E

on the same side of

BR

that

C

.

AD

intersect

\Gamma_1

in

P

and

CE

intersect

\Gamma_2

in

Q

. Prove that

APQC

is cyclic if and only if

AB=BC

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