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Parallelogram Leads to AQ = AR

Source: 2019 China TST Test 4 P1

April 9, 2019

geometrycircumcircleangle bisector

Problem Statement

Cyclic quadrilateral

ABCD

has circumcircle

(O)

. Points

M

and

N

are the midpoints of

BC

and

CD

, and

E

and

F

lie on

AB

and

AD

respectively such that

EF

passes through

O

and

EO=OF

. Let

EN

meet

FM

at

P

. Denote

S

as the circumcenter of

\triangle PEF

. Line

PO

intersects

AD

and

BA

at

Q

and

R

respectively. Suppose

OSPC

is a parallelogram. Prove that

AQ=AR

.

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