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1998 Croatia National Olympiad
Problem 1
trig equality in triangle
trig equality in triangle
Source: Croatia 1998 3rd Grade P1
June 8, 2021
trigonometry
geometry
Triangles
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the sides and
α
,
β
,
γ
\alpha,\beta,\gamma
α
,
β
,
γ
be the corresponding angles of a triangle. Prove the equality
(
b
c
+
c
b
)
cos
α
+
(
c
a
+
a
c
)
cos
β
+
(
a
b
+
b
a
)
cos
γ
=
3.
\left(\frac bc+\frac cb\right)\cos\alpha+\left(\frac ca+\frac ac\right)\cos\beta+\left(\frac ab+\frac ba\right)\cos\gamma=3.
(
c
b
+
b
c
)
cos
α
+
(
a
c
+
c
a
)
cos
β
+
(
b
a
+
a
b
)
cos
γ
=
3.
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