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National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2021 Sharygin Geometry Olympiad
4
4
Part of
2021 Sharygin Geometry Olympiad
Problems
(1)
Median perpendicular to a line
Source: XVII Sharygin Correspondence Round P4
3/2/2021
Let
A
B
C
D
ABCD
A
BC
D
be a square with center
O
O
O
, and
P
P
P
be a point on the minor arc
C
D
CD
C
D
of its circumcircle. The tangents from
P
P
P
to the incircle of the square meet
C
D
CD
C
D
at points
M
M
M
and
N
N
N
. The lines
P
M
PM
PM
and
P
N
PN
PN
meet segments
B
C
BC
BC
and
A
D
AD
A
D
respectively at points
Q
Q
Q
and
R
R
R
. Prove that the median of triangle
O
M
N
OMN
OMN
from
O
O
O
is perpendicular to the segment
Q
R
QR
QR
and equals to its half.
geometry
circumcircle