MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2024 Sharygin Geometry Olympiad
17
17
Part of
2024 Sharygin Geometry Olympiad
Problems
(1)
Beautiful problem with the incircle
Source: Sharygin Correspondence Round 2024 P17
3/6/2024
Let
A
B
C
ABC
A
BC
be a non-isosceles triangle,
ω
\omega
ω
be its incircle. Let
D
,
E
,
D, E,
D
,
E
,
and
F
F
F
be the points at which the incircle of
A
B
C
ABC
A
BC
touches the sides
B
C
,
C
A
,
BC, CA,
BC
,
C
A
,
and
A
B
AB
A
B
respectively. Let
M
M
M
be the point on ray
E
F
EF
EF
such that
E
M
=
A
B
EM = AB
EM
=
A
B
. Let
N
N
N
be the point on ray
F
E
FE
FE
such that
F
N
=
A
C
FN = AC
FN
=
A
C
. Let the circumcircles of
△
B
F
M
\triangle BFM
△
BFM
and
△
C
E
N
\triangle CEN
△
CEN
intersect
ω
\omega
ω
again at
S
S
S
and
T
T
T
respectively. Prove that
B
S
,
C
T
,
BS, CT,
BS
,
CT
,
and
A
D
AD
A
D
concur.
geometry
incircle