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2016 JBMO Shortlist
6
6
Part of
2016 JBMO Shortlist
Problems
(1)
2016 JBMO Shortlist G6
Source: 2016 JBMO Shortlist G6
10/8/2017
Given an acute triangle
A
B
C
{ABC}
A
BC
, erect triangles
A
B
D
{ABD}
A
B
D
and
A
C
E
{ACE}
A
CE
externally, so that
∠
A
D
B
=
∠
A
E
C
=
9
0
o
{\angle ADB= \angle AEC=90^o}
∠
A
D
B
=
∠
A
EC
=
9
0
o
and
∠
B
A
D
=
∠
C
A
E
{\angle BAD= \angle CAE}
∠
B
A
D
=
∠
C
A
E
. Let
A
1
∈
B
C
,
B
1
∈
A
C
{{A}_{1}}\in BC,{{B}_{1}}\in AC
A
1
∈
BC
,
B
1
∈
A
C
and
C
1
∈
A
B
{{C}_{1}}\in AB
C
1
∈
A
B
be the feet of the altitudes of the triangle
A
B
C
{ABC}
A
BC
, and let
K
K
K
and
K
,
L
{K,L}
K
,
L
be the midpoints of
[
B
C
1
]
[ B{{C}_{1}} ]
[
B
C
1
]
and
B
C
1
,
C
B
1
{BC_1, CB_1}
B
C
1
,
C
B
1
, respectively. Prove that the circumcenters of the triangles
A
K
L
,
A
1
B
1
C
1
AKL,{{A}_{1}}{{B}_{1}}{{C}_{1}}
A
K
L
,
A
1
B
1
C
1
and
D
E
A
1
{DEA_1}
D
E
A
1
are collinear.(Bulgaria)
geometry
JBMO