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Contests
National and Regional Contests
Vietnam Contests
VMEO = Vietnam Mathematical E-Olympiad
VMEO III 2006 Shortlist
G2
G2
Part of
VMEO III 2006 Shortlist
Problems
(1)
Hard geo
Source: VMEO III
3/30/2016
Given a triangle
A
B
C
ABC
A
BC
, incircle
(
I
)
(I)
(
I
)
touches
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D,E,F
D
,
E
,
F
respectively. Let
M
M
M
be a point inside
A
B
C
ABC
A
BC
. Prove that
M
M
M
lie on
(
I
)
(I)
(
I
)
if and only if one number among
A
E
⋅
S
B
M
C
,
B
F
⋅
S
C
M
A
,
C
D
⋅
S
A
M
B
\sqrt{AE\cdot S_{BMC}},\sqrt{BF\cdot S_{CMA}},\sqrt{CD\cdot S_{AMB}}
A
E
⋅
S
BMC
,
BF
⋅
S
CM
A
,
C
D
⋅
S
A
MB
is sum of two remaining numbers (
S
A
B
C
S_{ABC}
S
A
BC
denotes the area of triangle
A
B
C
ABC
A
BC
)
geometry