Arithmetic, Geometry, Harmony, and Gold

I recently ran across a theorem connecting the arithmetic mean, geometric mean, harmonic mean, and the golden ratio. Each of these comes fairly often, and there are elegant connections between them, but I don’t recall seeing all four together in one theorem before.

Here’s the theorem [1]:

The arithmetic, geometric, and harmonic means of two positive real numbers are the lengths of the sides of a right triangle if, and only if, the ratio of the arithmetic to the harmonic mean is the Golden Ratio.

The proof given in [1] is a straight-forward calculation, only slightly longer than the statement of the theorem.

The conclusion of the theorem stops short of saying how to construct the triangle, though this is a simple exercise, which we carry out here.

Given two positive numbers, a and b, the three means are defined as follows.

AM = (a + b)/2 GM = √ab HM = 2ab/(a + b)

Denote the Golden Ratio by

φ = (1 + √5)/2.

Then the equation AM/HM = φ is equivalent to the quadratic equation

a² + (2 − 4φ)ab + b² = 0.

The means are all homogeneous functions of a and b, i.e. if we multiply a and b by a constant, we multiply the three means by the same constant. Therefore we can set one of the parameters to 1 without loss of generality. Setting b = 1 gives

a² + (2 − 4φ)a + 1 = 0

and so there are two solutions:

a = 2φ − 3

and

a = 2φ + 1.

However, there is in a sense only one solution: the two solutions are reciprocals of each other, reversing the roles of a and b. So while there are two solutions to the quadratic equation, there is only one triangle, up to similarity.

[1] Angelo Di Domenico. The Golden Ratio: The Right Triangle: And the Arithmetic, Geometric, and Harmonic Means. The Mathematical Gazette Vol. 89, No. 515 (July, 2005), p. 261

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