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Orthogonal lines meeting on a circle formed by midpoints

Source: 2021 Macedonian Team Selection Test P2

May 30, 2021

geometry

Problem Statement

Let

ABC

be an acute triangle such that

AB<AC

. Denote by

A'

the reflection of

A

with respect to

BC

. The circumcircle of

A'BC

meets the rays

AB

and

AC

at

D

and

E

respectively, such that

B

is between

A

and

D

, and

E

is between

A

and

C

. Denote by

P

and

Q

the midpoints of the segments

CD

and

BE

, and let

S

be the midpoint of

BC

. Show that the lines

BC

and

AA'

meet on the circumcircle of

PQS

.
Proposed by Nikola Velov

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