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Benelux
2019 Benelux
3
3
Part of
2019 Benelux
Problems
(1)
Bisector passes through circumcenter
Source: Benelux MO 2019 P3
4/28/2019
Two circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
intersect at points
A
A
A
and
Z
Z
Z
(with
A
≠
Z
A\neq Z
A
=
Z
). Let
B
B
B
be the centre of
Γ
1
\Gamma_1
Γ
1
and let
C
C
C
be the centre of
Γ
2
\Gamma_2
Γ
2
. The exterior angle bisector of
∠
B
A
C
\angle{BAC}
∠
B
A
C
intersects
Γ
1
\Gamma_1
Γ
1
again at
X
X
X
and
Γ
2
\Gamma_2
Γ
2
again at
Y
Y
Y
. Prove that the interior angle bisector of
∠
B
Z
C
\angle{BZC}
∠
BZC
passes through the circumcenter of
△
X
Y
Z
\triangle{XYZ}
△
X
Y
Z
.For points
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
that lie on a line
ℓ
\ell
ℓ
in that order, and a point
S
S
S
not on
ℓ
\ell
ℓ
, the interior angle bisector of
∠
P
Q
S
\angle{PQS}
∠
PQS
is the line that divides
∠
P
Q
S
\angle{PQS}
∠
PQS
into two equal angles, while the exterior angle bisector of
∠
P
Q
S
\angle{PQS}
∠
PQS
is the line that divides
∠
R
Q
S
\angle{RQS}
∠
RQS
into two equal angles.
geometry
circumcircle