Arithmetic, Geometric and Harmonic Mean

 by

   Markus Heisss

 Würzburg, Bavaria

 2018

 

The copying of the following graphics is allowed, but without changes.

 [To get a bigger picture, please click it with the cursor.]

 

First to the common formulas:

 

Graphic by Heisss, Würzburg, Germany
Fig. 1: Common formulas of the arithmetic, geometric and harmonic mean

 

For the rest of this website we always take the means only of two values!

 

arithmetic, geometric and harmonic mean, comparison, formula
Fig. 2: Construction of the means of two values

 

In the graphic we see, that ...

cos φ = BC/AB and cos φ = CD/BC.

From both equations follows:  BC² = AB CD or ...

by Markus Heiss, Würzburg

 

Let's take again the arithmetic and harmonic mean

 from the arithmetic and harmonic mean! We get:

geometric harmonic arithmetic mean

 

We could continue this. And it follows:

 

geometric mean formula, Graphic by Heisss

 

Astonishingly, the n in this formula can be negative!

[More informationon on that you find here.]

 

An interesting application of the geometric mean is the

'Theorem of Three Shapes'.

You will find it [here].

 

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Are you interested in my other geometrical discoveries?

[here]