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International Contests
Caucasus Mathematical Olympiad
2022 Caucasus Mathematical Olympiad
7
7
Part of
2022 Caucasus Mathematical Olympiad
Problems
(1)
Variable line touches a fixed circle
Source: VII Caucasus Mathematical Olympiad
3/13/2022
Point
P
P
P
is chosen on the leg
C
B
CB
CB
of right triangle
A
B
C
ABC
A
BC
(
∠
A
C
B
=
9
0
∘
\angle ACB = 90^\circ
∠
A
CB
=
9
0
∘
). The line
A
P
AP
A
P
intersects the circumcircle of
A
B
C
ABC
A
BC
at point
Q
Q
Q
. Let
L
L
L
be the midpoint of
P
B
PB
PB
. Prove that
Q
L
QL
Q
L
is tangent to a fixed circle independent of the choice of point
P
P
P
.
geometry