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IGO 2022 advanced/free P5

Source: Iranian Geometry Olympiad 2022 P5 Advanced, Free

December 13, 2022

geometry

Problem Statement

Let

ABC

be an acute triangle inscribed in a circle

\omega

with center

O

. Points

E

,

F

lie on its side

AC

,

AB

, respectively, such that

O

lies on

EF

and

BCEF

is cyclic. Let

R

,

S

be the intersections of

EF

with the shorter arcs

AB

,

AC

of

\omega

, respectively. Suppose

K

,

L

are the reflection of

R

about

C

and the reflection of

S

about

B

, respectively. Suppose that points

P

and

Q

lie on the lines

BS

and

RC

, respectively, such that

PK

and

QL

are perpendicular to

BC

. Prove that the circle with center

P

and radius

PK

is tangent to the circumcircle of

RCE

if and only if the circle with center

Q

and radius

QL

is tangent to the circumcircle of

BFS

.
Proposed by Mehran Talaei

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