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An interesting geometry

Source: All-Russian Olympiad 2019 grade 10 problem 4

April 23, 2019

geometrycircumcirclegeometric transformationreflection

Problem Statement

Let

ABC

be an acute-angled triangle with

AC<BC.

A circle passes through

A

and

B

and crosses the segments

AC

and

BC

again at

A_1

and

B_1

respectively. The circumcircles of

A_1B_1C

and

ABC

meet each other at points

P

and

C.

The segments

S

and

A_1B

intersect at

S.

Let

Q

and

C

be the reflections of

S

in the lines

CA

and

CB

respectively. Prove that the points

P,

Q,

R,

and

C

are concyclic.

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