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Problems
Contests
National and Regional Contests
Nepal Contests
Nepal TST
2024 Nepal TST
P5
P5
Part of
2024 Nepal TST
Problems
(1)
Angle geometry
Source: 2024 Nepal TST P5
4/12/2024
Let
A
B
C
ABC
A
BC
be an acute triangle so that
2
B
C
=
A
B
+
A
C
,
2BC = AB + AC,
2
BC
=
A
B
+
A
C
,
with incenter
I
.
I{}.
I
.
Let
A
I
AI{}
A
I
meet
B
C
BC{}
BC
at point
A
′
.
A'.{}
A
′
.
The perpendicular bisector of
A
A
′
AA'{}
A
A
′
meets
B
I
BI{}
B
I
and
C
I
CI{}
C
I
at points
B
′
B'{}
B
′
and
C
′
C'{}
C
′
respectively. Let
A
B
′
AB'{}
A
B
′
intersect
(
A
B
C
)
(ABC)
(
A
BC
)
at
X
X{}
X
and let
X
I
XI{}
X
I
intersect
A
C
′
AC'{}
A
C
′
at
X
′
.
X'{}.
X
′
.
Prove that
2
∠
X
X
′
A
′
=
∠
A
B
C
.
2\angle XX'A'=\angle ABC.{}
2∠
X
X
′
A
′
=
∠
A
BC
.
(Proposed by Kang Taeyoung, South Korea)
geometry
angles