\ Lorian Saceanu's Inequality for All Triangles

Lorian Saceanu's Inequality for All Triangles

Problem

Lorian Saceanu's Inequality for All Triangles, problem

Solution

We know that

$\sin 2A +\sin 2B +\sin 2C = 4\sin A\sin B\sin C$

which reduces the required inequality to

$\sin A\sin B\sin C\ge \sin 2A\sin 2B\sin 2C.$

The latter, through the double angle formulas, is equivalent to

$\displaystyle \frac{1}{8}\ge\cos A\cos B\cos C,$

a well known inequality.

Acknowledgment

This problem was shared by Lorian Saceanu at the Easy Beautiful Math Facebook group where there's an additional solution. Originally, the problem was formulated for acute triangle and later declared true for all triangles. At the link there are additional proofs.

 

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