# Reeb vector field

In mathematics, **Weitzenböck's inequality**, named after Roland Weitzenböck, states that for a triangle of side lengths $a$, $b$, $c$, and area $\mathrm{\Delta}$, the following inequality holds:

Equality occurs if and only if the triangle is equilateral. Pedoe's inequality is a generalization of Weitzenböck's inequality.

## Proofs

The proof of this inequality was set as a question in the International Mathematical Olympiad of 1961. Even so, the result is not too difficult to derive using Heron's formula for the area of a triangle:

### First method

This method assumes no knowledge of inequalities except that all squares are nonnegative.

and the result follows immediately by taking the positive square root of both sides. From the first inequality we can also see that equality occurs only when $a=b=c$ and the triangle is equilateral.

### Second method

This proof assumes knowledge of the rearrangement inequality and the arithmetic-geometric mean inequality.

As we have used the rearrangement inequality and the arithmetic-geometric mean inequality, equality only occurs when $a=b=c$ and the triangle is equilateral.

### Third method

It can be shown that the area of the inner Napoleon's triangle is:

and therefore greater than or equal to 0.

## External links

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