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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1976 Vietnam National Olympiad
2
2
Part of
1976 Vietnam National Olympiad
Problems
(1)
(acosA + bcosB + ccosC)/(asinA + bsinB + csinC) = (a + b + c)/9R
Source: Vietnamese MO (VMO) 1976 P2
8/20/2018
Find all triangles
A
B
C
ABC
A
BC
such that
a
c
o
s
A
+
b
c
o
s
B
+
c
c
o
s
C
a
s
i
n
A
+
b
s
i
n
B
+
c
s
i
n
C
=
a
+
b
+
c
9
R
\frac{a cos A + b cos B + c cos C}{a sin A + b sin B + c sin C} =\frac{a + b + c}{9R}
a
s
in
A
+
b
s
in
B
+
cs
in
C
a
cos
A
+
b
cos
B
+
ccos
C
=
9
R
a
+
b
+
c
, where, as usual,
a
,
b
,
c
a, b, c
a
,
b
,
c
are the lengths of sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
and
R
R
R
is the circumradius.
geometry
trigonometry
circumradius