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JBMO ShortLists
2004 JBMO Shortlist
4
4
Part of
2004 JBMO Shortlist
Problems
(1)
4 semicircles inside a right triangle
Source: JBMO Shortlist 2004
10/13/2017
Let
A
B
C
ABC
A
BC
be a triangle with
m
(
∠
C
)
=
9
0
∘
m (\angle C) = 90^\circ
m
(
∠
C
)
=
9
0
∘
and the points
D
∈
[
A
C
]
,
E
∈
[
B
C
]
D \in [AC], E\in [BC]
D
∈
[
A
C
]
,
E
∈
[
BC
]
. Inside the triangle we construct the semicircles
C
1
,
C
2
,
C
3
,
C
4
C_1, C_2, C_3, C_4
C
1
,
C
2
,
C
3
,
C
4
of diameters
[
A
C
]
,
[
B
C
]
,
[
C
D
]
,
[
C
E
]
[AC], [BC], [CD], [CE]
[
A
C
]
,
[
BC
]
,
[
C
D
]
,
[
CE
]
and let
{
C
,
K
}
=
C
1
∩
C
2
,
{
C
,
M
}
=
C
3
∩
C
4
,
{
C
,
L
}
=
C
2
∩
C
3
,
{
C
,
N
}
=
C
1
∩
C
4
\{C, K\} = C_1 \cap C_2, \{C, M\} =C_3 \cap C_4, \{C, L\} = C_2 \cap C_3, \{C, N\} =C_1 \cap C_4
{
C
,
K
}
=
C
1
∩
C
2
,
{
C
,
M
}
=
C
3
∩
C
4
,
{
C
,
L
}
=
C
2
∩
C
3
,
{
C
,
N
}
=
C
1
∩
C
4
. Show that points
K
,
L
,
M
,
N
K, L, M, N
K
,
L
,
M
,
N
are concyclic.
JBMO
geometry