1

Part of 1965 Polish MO Finals

Problems(1)

\pi /3 <= ( a A+ b B+c C}{a+b+c} < \pi / 2

Source: Polish MO Finals 1965 p1

Prove the theorem:
the lengths

a

b

c

of the sides of a triangle and the arc measures

\alpha

\beta

\gamma

of its opposite angles satisfy the inequalities

\frac{\pi}{3}\leq \frac{a \alpha + b \beta +c \gamma}{a+b+c}<\frac{\pi }{ 2}.

algebrainequalitiesgeometryGeometric Inequalities