The Inequality of Arithmetic and Geometric Means from Multiple Perspectives

Richard Askey, Ryota Matsuura, and Sarah Sword (PDF):

Given three numbers a, b, and c, we can find their mean (or average) as (a + b + c)/3. More precisely, this expression yields the
arithmetic mean of a, b, and c. A different kind of mean, however, uses the product of these numbers instead of their sum. It is called the geometric mean and is given by the expression (abc)1/3. We may interpret the geometric mean of nonnegative a, b, and c as the side length of a cube whose volume is the same as that of a right rectangular prism with dimensions a, b, and c.
In NCTM’s Focus in High School Mathematics: Reasoning and Sense Making in Algebra (2010), Graham, Cuoco, and Zimmermann use similar triangles and angles inscribed in circles to prove that, for two nonnegative numbers a and b, the arithmetic mean is greater than or equal to the geo- metric mean. Symbolically, this is written as (a + b)/2 ≥ (ab)1/2. This fact, called the inequality of arithmetic and geometric means (AGM), is actu- ally true for any number of nonnegative values. For three nonnegative numbers a, b, and c, for instance, we have (a + b + c)/3 ≥ (abc)1/3.