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National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2024 Sharygin Geometry Olympiad
3
3
Part of
2024 Sharygin Geometry Olympiad
Problems
(1)
Sharygin 2024 Correspondence P3
Source: Sharygin Correspondence Round 2024 P3
3/6/2024
Let
A
B
C
ABC
A
BC
be an acute-angled triangle, and
M
M
M
be the midpoint of the minor arc
B
C
BC
BC
of its circumcircle. A circle
ω
\omega
ω
touches the side
A
B
,
A
C
AB, AC
A
B
,
A
C
at points
P
,
Q
P, Q
P
,
Q
respectively and passes through
M
M
M
. Prove that
B
P
+
C
Q
=
P
Q
BP + CQ = PQ
BP
+
CQ
=
PQ
.
geometry