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ASU 318 All Soviet Union MO 1981 P/2 < p < 3P/4 perimeter inequality

Source:

July 23, 2019

geometryperimeterinequalitiesgeometric inequalityratio

Problem Statement

The points

C_1, A_1, B_1

belong to

[AB], [BC], [CA]

sides, respectively, of the triangle

ABC

\frac{|AC_1|}{|C_1B| }=\frac{ |BA_1|}{|A_1C| }= \frac{|CB_1|}{|B_1A| }= \frac{1}{3}

Prove that the perimeter

P

of the triangle

ABC

and the perimeter

p

of the triangle

A_1B_1C_1

, satisfy inequality

\frac{P}{2} < p < \frac{3P}{4}