Prove that the following inequality holds in any triangle:

**Solution**

Let denote the semiperimeter of the triangle. Using the cosine theorem we have that

from which it follows that

(1)

(2)

(3)

Thus, by Cauchy’s inequality we have:

Hence,

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Prove that the following inequality holds in any triangle:

**Solution**

Let denote the semiperimeter of the triangle. Using the cosine theorem we have that

from which it follows that

(1)

(2)

(3)

Thus, by Cauchy’s inequality we have:

Hence,

Remarks1.We have:2.We haveand

from which it follows

where is the inradius and the circumradius.