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2022 Benelux
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Part of
2022 Benelux
Problems
(1)
Four concyclic points (BxMO 2022, Problem 3)
Source: BxMO 2022, Problem 3
5/1/2022
Let
A
B
C
ABC
A
BC
be a scalene acute triangle. Let
B
1
B_1
B
1
be the point on ray
[
A
C
[AC
[
A
C
such that
∣
A
B
1
∣
=
∣
B
B
1
∣
|AB_1|=|BB_1|
∣
A
B
1
∣
=
∣
B
B
1
∣
. Let
C
1
C_1
C
1
be the point on ray
[
A
B
[AB
[
A
B
such that
∣
A
C
1
∣
=
∣
C
C
1
∣
|AC_1|=|CC_1|
∣
A
C
1
∣
=
∣
C
C
1
∣
. Let
B
2
B_2
B
2
and
C
2
C_2
C
2
be the points on line
B
C
BC
BC
such that
∣
A
B
2
∣
=
∣
C
B
2
∣
|AB_2|=|CB_2|
∣
A
B
2
∣
=
∣
C
B
2
∣
and
∣
B
C
2
∣
=
∣
A
C
2
∣
|BC_2|=|AC_2|
∣
B
C
2
∣
=
∣
A
C
2
∣
. Prove that
B
1
B_1
B
1
,
C
1
C_1
C
1
,
B
2
B_2
B
2
,
C
2
C_2
C
2
are concyclic.
geometry
BxMO