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4 concyclic points

Source: 2019 MEMO Problem I-3

August 29, 2019

geometrycircumcircleperpendicular bisectorMEMO 2019memomoving points

Problem Statement

Let

ABC

be an acute-angled triangle with

AC>BC

and circumcircle

\omega

. Suppose that

P

is a point on

\omega

such that

AP=AC

and that

P

is an interior point on the shorter arc

BC

of

\omega

. Let

Q

be the intersection point of the lines

AP

and

BC

. Furthermore, suppose that

R

is a point on

\omega

such that

QA=QR

and

R

is an interior point of the shorter arc

AC

of

\omega

. Finally, let

S

be the point of intersection of the line

BC

with the perpendicular bisector of the side

AB

. Prove that the points

P, Q, R

and

S

are concyclic.
Proposed by Patrik Bak, Slovakia

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