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National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2021 Sharygin Geometry Olympiad
17
17
Part of
2021 Sharygin Geometry Olympiad
Problems
(1)
An incenter config with a beautiful condition
Source: XVII Sharygin Correspondence Round P17
3/2/2021
Let
A
B
C
ABC
A
BC
be an acute-angled triangle. Points
A
0
A_0
A
0
and
C
0
C_0
C
0
are the midpoints of minor arcs
B
C
BC
BC
and
A
B
AB
A
B
respectively. A circle passing though
A
0
A_0
A
0
and
C
0
C_0
C
0
meet
A
B
AB
A
B
and
B
C
BC
BC
at points
P
P
P
and
S
S
S
,
Q
Q
Q
and
R
R
R
respectively (all these points are distinct). It is known that
P
Q
∥
A
C
PQ\parallel AC
PQ
∥
A
C
. Prove that
A
0
P
+
C
0
S
=
C
0
Q
+
A
0
R
A_0P+C_0S=C_0Q+A_0R
A
0
P
+
C
0
S
=
C
0
Q
+
A
0
R
.
geometry
incenter