MathDB
Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Kyiv City MO
Kyiv City MO - geometry
Kyiv City MO Seniors Round2 2010+ geometry
2021.11.3.1
2021.11.3.1
Part of
Kyiv City MO Seniors Round2 2010+ geometry
Problems
(1)
product of radii, common tangents to circles (2021 Kyiv City MO Round2 11.3.1)
Source:
2/15/2021
Two circles
k
1
k_1
k
1
and
k
2
k_2
k
2
with radii
r
1
r_1
r
1
and
r
2
r_2
r
2
have no common points. The line
A
B
AB
A
B
is a common internal tangent, and the line
C
D
CD
C
D
is a common external tangent to these circles, where
A
,
C
∈
k
1
A, C \in k_1
A
,
C
∈
k
1
and
B
,
D
∈
k
2
B, D \in k_2
B
,
D
∈
k
2
. Knowing that
A
B
=
12
AB=12
A
B
=
12
and
C
D
=
16
CD =16
C
D
=
16
, find the value of the product
r
1
r
2
r_1r_2
r
1
r
2
.
geometry
Tangents
circles