## Inequality in triangle

We denote by the points where the incircle of triangle touches the sides , respectively. Prove that .

**Solution:** Denote by . Then , where is the radius if the incircle. We know that , where is the area of the given triangle. Then, by Heron’s formula we get .

Replacing we get that our inequality is equivalent to the following:

.

Use the AM-GM inequality to prove that the second part is greater than .

For the other part, prove that and the other permutations, using the GM-HM inequality.

**Remark:** For we have:

.

The first one is called AM-GM inequality, and the second one inequality.

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Categories: Geometry, Inequalities, Olympiad, Problem Solving
Inequalities

There is more geometrical solution. By Cauchy inequality:

Which is equivalent to . This is true, since it is kown that and .

I guess that you denote by the area of the triangle . Thank you for this very nice and geometrical solution.