MathDB
Problems
Contests
International Contests
Kvant Problems
Kvant 2023
M2773
M2773
Part of
Kvant 2023
Problems
(1)
Variable chord tangent to a circle
Source: Kvant Magazine No. 11-12 2023 M2773
3/16/2024
The circle
ω
\omega
ω
lies inside the circle
Ω
\Omega
Ω
and touches it internally at
T
.
T.
T
.
Let
X
Y
XY{}
X
Y
be a variable chord of the circle
Ω
\Omega
Ω
touching
ω
.
\omega.
ω
.
Denote by
X
′
X'
X
′
and
Y
′
Y'
Y
′
the midpoints of the arcs
T
Y
TY{}
T
Y
and
T
X
TX{}
TX
which do not contain
X
X{}
X
and
Y
Y{}
Y
respectively. Prove that all possible lines
X
′
Y
′
X'Y'
X
′
Y
′
pass through a fixed point.Proposed by F. Petrov
geometry
Fixed point