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National and Regional Contests
North Macedonia Contests
Macedonian Team Selection Test
2021 Macedonian Team Selection Test
Problem 2
Problem 2
Part of
2021 Macedonian Team Selection Test
Problems
(1)
Orthogonal lines meeting on a circle formed by midpoints
Source: 2021 Macedonian Team Selection Test P2
5/30/2021
Let
A
B
C
ABC
A
BC
be an acute triangle such that
A
B
<
A
C
AB<AC
A
B
<
A
C
. Denote by
A
′
A'
A
′
the reflection of
A
A
A
with respect to
B
C
BC
BC
. The circumcircle of
A
′
B
C
A'BC
A
′
BC
meets the rays
A
B
AB
A
B
and
A
C
AC
A
C
at
D
D
D
and
E
E
E
respectively, such that
B
B
B
is between
A
A
A
and
D
D
D
, and
E
E
E
is between
A
A
A
and
C
C
C
. Denote by
P
P
P
and
Q
Q
Q
the midpoints of the segments
C
D
CD
C
D
and
B
E
BE
BE
, and let
S
S
S
be the midpoint of
B
C
BC
BC
. Show that the lines
B
C
BC
BC
and
A
A
′
AA'
A
A
′
meet on the circumcircle of
P
Q
S
PQS
PQS
.Proposed by Nikola Velov
geometry