# Definition:Heronian Mean

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## Definition

The **Heronian mean** of two numbers $x$ and $y$ is defined as:

- $H = \dfrac {x + \sqrt {x y} + y} 3$

It can also be defined as:

- $H = \dfrac 2 3 \paren {\dfrac {x + y} 2} + \dfrac 1 3 \sqrt {x y}$

Thus it is seen to be a weighted mean of their arithmetic mean and geometric mean.

## Source of Name

This entry was named for Heron of Alexandria.