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JBMO ShortLists
2004 JBMO Shortlist
3
3
Part of
2004 JBMO Shortlist
Problems
(1)
3 circles tangent internally to circumcircle and sides ABC
Source: JBMO Shortlist 2004
10/13/2017
Let
A
B
C
ABC
A
BC
be a triangle inscribed in circle
C
C
C
. Circles
C
1
,
C
2
,
C
3
C_1, C_2, C_3
C
1
,
C
2
,
C
3
are tangent internally with circle
C
C
C
in
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
and tangent to sides
[
B
C
]
,
[
C
A
]
,
[
A
B
]
[BC], [CA], [AB]
[
BC
]
,
[
C
A
]
,
[
A
B
]
in points
A
2
,
B
2
,
C
2
A_2, B_2, C_2
A
2
,
B
2
,
C
2
respectively, so that
A
,
A
1
A, A_1
A
,
A
1
are on one side of
B
C
BC
BC
and so on. Lines
A
1
A
2
,
B
1
B
2
A_1A_2, B_1B_2
A
1
A
2
,
B
1
B
2
and
C
1
C
2
C_1C_2
C
1
C
2
intersect the circle
C
C
C
for second time at points
A
’
,
B
’
A’,B’
A
’
,
B
’
and
C
’
C’
C
’
, respectively. If
M
=
B
B
’
∩
C
C
’
M = BB’ \cap CC’
M
=
BB
’
∩
CC
’
, prove that
m
(
∠
M
A
A
’
)
=
9
0
∘
m (\angle MAA’) = 90^\circ
m
(
∠
M
AA
’
)
=
9
0
∘
.
geometry
circumcircle
JBMO