MathDB
Problems
Contests
International Contests
Kvant Problems
Kvant 2020
M2606
M2606
Part of
Kvant 2020
Problems
(1)
Geometry with three circles
Source: Kvant Magazine No. 6 2020 M2606
3/9/2023
Three circles
ω
1
,
ω
2
\omega_1,\omega_2
ω
1
,
ω
2
and
ω
3
\omega_3
ω
3
pass through one point
D
D{}
D
. Let
A
A{}
A
be the intersection of
ω
1
\omega_1
ω
1
and
ω
3
\omega_3
ω
3
, and
E
E{}
E
be the intersections of
ω
3
\omega_3
ω
3
and
ω
2
\omega_2
ω
2
and
F
F{}
F
be the intersection of
ω
2
\omega_2
ω
2
and
ω
1
\omega_1
ω
1
. It is known that
ω
3
\omega_3
ω
3
passes through the center
B
B{}
B
of the circle
ω
2
\omega_2
ω
2
. The line
E
F
EF
EF
intersects
ω
1
\omega_1
ω
1
a second time at the point
G
G{}
G
. Prove that
∠
G
A
B
=
9
0
∘
\angle GAB=90^\circ
∠
G
A
B
=
9
0
∘
.Proposed by K. Knop
geometry
Kvant