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National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2021 Sharygin Geometry Olympiad
7
7
Part of
2021 Sharygin Geometry Olympiad
Problems
(1)
A line through incenter bisects a segment.
Source: XVII Sharygin Correspondence Round P7
3/2/2021
The incircle of triangle
A
B
C
ABC
A
BC
centered at
I
I
I
touches
C
A
,
A
B
CA,AB
C
A
,
A
B
at points
E
,
F
E,F
E
,
F
respectively. Let points
M
,
N
M,N
M
,
N
of line
E
F
EF
EF
be such that
C
M
=
C
E
CM=CE
CM
=
CE
and
B
N
=
B
F
BN=BF
BN
=
BF
. Lines
B
M
BM
BM
and
C
N
CN
CN
meet at point
P
P
P
. Prove that
P
I
PI
P
I
bisects segment
M
N
MN
MN
.
geometry
incenter
bisects segment