Arithmetic mean and geometric mean

An elegant result that also has a beautiful geometric interpretation on the circle: the arithmetic mean of two positive numbers is always at least as large as their geometric mean.

Theorem — Inequality between the means

The arithmetic mean of two positive numbers is always $\ge$ the geometric mean: $\frac{a+b}{2} \ge \sqrt{ab}, \qquad \forall\, a,b > 0.$ Equality holds if and only if $a=b$.

The geometric interpretation: on a semicircle of diameter $a+b$, the radius $\frac{a+b}{2}$ (which runs from the centre to a point on the circle) is always greater than or equal to the height $\sqrt{ab}$ relative to the hypotenuse.

The height $CH=\sqrt{ab}$ never exceeds the radius $CO=\frac{a+b}{2}$.

Links

Topics: Euclidean circle
Concepts: Arithmetic mean · Geometric mean
Skills: Proving