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International Contests
Silk Road
2021 Silk Road
3
3
Part of
2021 Silk Road
Problems
(1)
midpoints and circumcircles
Source: 2021 SRMC P3
6/28/2021
In a triangle
A
B
C
ABC
A
BC
,
M
M
M
is the midpoint of the
A
B
AB
A
B
. A point
B
1
B_1
B
1
is marked on
A
C
AC
A
C
such that
C
B
=
C
B
1
CB=CB_1
CB
=
C
B
1
. Circle
ω
\omega
ω
and
ω
1
\omega_1
ω
1
, the circumcircles of triangles
A
B
C
ABC
A
BC
and
B
M
B
1
BMB_1
BM
B
1
, respectively, intersect again at
K
K
K
. Let
Q
Q
Q
be the midpoint of the arc
A
C
B
ACB
A
CB
on
ω
\omega
ω
. Let
B
1
Q
B_1Q
B
1
Q
and
B
C
BC
BC
intersect at
E
E
E
. Prove that
K
C
KC
K
C
bisects
B
1
E
B_1E
B
1
E
.M. Kungozhin
geometry