I checked my mailbox and found a post which was buried long time ago.
It was about a geometry inequality, and it is really interesting.
where the symbol represents cyclic sum.
The proof was elementary now. But I still like this, full of magic skills.
First we introduce two formulae of triangular geometry.
where the symbols’ meanings are obvious, I suppose.
Then the inequality turns into:
however, we may prove a simpler result by using:
Then we have to prove the function:
Fortunately, we can use Jensen Inequality here, we want to prove:
which means we need find the maximum of the RHS.
We set, , then,
Thus, is a convex function.
It is proven.
It is a very strong inequality actually.