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.

**Proof:**

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.

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