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Kvant 2021
M2643
M2643
Part of
Kvant 2021
Problems
(1)
Internal tangent circles
Source: Kvant Magazine No. 3 2021 M2643
3/9/2023
The circles
ω
\omega
ω
and
Ω
\Omega
Ω
touch each other internally at
A
A{}
A
. In a larger circle
Ω
\Omega
Ω
consider the chord
C
D
CD
C
D
which touches
ω
\omega
ω
at
B
B{}
B
. It is known that the chord
A
B
AB
A
B
is not a diameter of
ω
\omega
ω
. The point
M
M{}
M
is the middle of the segment
A
B
AB{}
A
B
. Prove that the circumcircle of the triangle
C
M
D
CMD
CM
D
passes through the center of
ω
\omega
ω
.Proposed by P. Bibikov
geometry
Kvant