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Bulgaria Contests
Bulgarian Spring Mathematical Competition
2024 Bulgarian Spring Mathematical Competition
10.2
10.2
Part of
2024 Bulgarian Spring Mathematical Competition
Problems
(1)
Varying circle through a vertex and the incenter and a fixed angle
Source: Bulgarian Spring Tournament 2024 10.2
3/31/2024
Let
A
B
C
ABC
A
BC
be a triangle and a circle
ω
\omega
ω
through
C
C
C
and its incenter
I
I
I
meets
C
A
,
C
B
CA, CB
C
A
,
CB
at
P
,
Q
P, Q
P
,
Q
. The circumcircles
(
C
P
Q
)
(CPQ)
(
CPQ
)
and
(
A
B
C
)
(ABC)
(
A
BC
)
meet at
L
L
L
. The angle bisector of
∠
A
L
B
\angle ALB
∠
A
L
B
meets
A
B
AB
A
B
at
K
K
K
. Show that, as
ω
\omega
ω
varies,
∠
P
K
Q
\angle PKQ
∠
P
K
Q
is constant.
geometry