MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2019 Sharygin Geometry Olympiad
17
17
Part of
2019 Sharygin Geometry Olympiad
Problems
(1)
Sharygin CR 2019 P17
Source: Sharygin CR 2019 P17 (Grade 10 - 11)
3/6/2019
Three circles
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
,
ω
3
\omega_3
ω
3
are given. Let
A
0
A_0
A
0
and
A
1
A_1
A
1
be the common points of
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
,
B
0
B_0
B
0
and
B
1
B_1
B
1
be the common points of
ω
2
\omega_2
ω
2
and
ω
3
\omega_3
ω
3
,
C
0
C_0
C
0
and
C
1
C_1
C
1
be the common points of
ω
3
\omega_3
ω
3
and
ω
1
\omega_1
ω
1
. Let
O
i
,
j
,
k
O_{i,j,k}
O
i
,
j
,
k
be the circumcenter of triangle
A
i
B
j
C
k
A_iB_jC_k
A
i
B
j
C
k
. Prove that the four lines of the form
O
i
j
k
O
1
−
i
,
1
−
j
,
1
−
k
O_{ijk}O_{1 - i,1 - j,1 - k}
O
ijk
O
1
−
i
,
1
−
j
,
1
−
k
are concurrent or parallel.
geometry