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,
Remarks
1. We have:
2. We have
and
from which it follows
where
is the inradius and
the circumradius.