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 = 2*ab*/(*a* + *b*)

Denote the Golden Ratio by

φ = (1 + √5)/2.

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

*a*² + (2 − 4φ)*ab* + *b*² = 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

*a*² + (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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