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

Source: Polish MO Finals 1965 p1

August 30, 2024
algebrainequalitiesgeometryGeometric Inequalities

Problem Statement

Prove the theorem:
the lengths a a, b b , c c of the sides of a triangle and the arc measures α \alpha , β \beta , γ \gamma of its opposite angles satisfy the inequalities π3≤aα+bβ+cγa+b+c<π2.\frac{\pi}{3}\leq \frac{a \alpha + b \beta +c \gamma}{a+b+c}<\frac{\pi }{ 2}.
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