contraharmonic proportion
Three positive numbers $x$, $m$, $y$ are in contraharmonic proportion, if the ratio of the difference of the second and the first number to the difference of the third and the second number is equal the ratio of the third and the first number, i.e. if

$\frac{mx}{ym}}={\displaystyle \frac{y}{x}}.$ 

(1) 
The middle number $m$ is then called the contraharmonic mean (sometimes antiharmonic mean) of the first and the last number.
The contraharmonic proportion has very probably been known in the proportion doctrine of the Pythagoreans, since they have in a manner to (1) described the classical Babylonian means:

$$\frac{mx}{ym}=\frac{m}{m}\mathit{\hspace{1em}\hspace{1em}}(\text{arithmetic mean}m)$$ 


$$\frac{mx}{ym}=\frac{m}{y}\mathit{\hspace{1em}\hspace{1em}}(\text{geometric mean}m)$$ 


$$\frac{mx}{ym}=\frac{x}{y}\mathit{\hspace{1em}\hspace{1em}}(\text{harmonic mean}m)$$ 

The contraharmonic mean $m$ is between $x$ and $y$. Indeed, if we solve it from (1), we get

$m={\displaystyle \frac{{x}^{2}+{y}^{2}}{x+y}},$ 

(2) 
and if we assume that $x\leqq y$, we see that

$$x=\frac{{x}^{2}+xy}{x+y}\leqq \frac{{x}^{2}+{y}^{2}}{x+y}\leqq \frac{xy+{y}^{2}}{x+y}=y.$$ 

The contraharmonic mean $c$ is the greatest of all the mentioned means,

$$x\leqq h\leqq g\leqq a\leqq c\leqq y,$$ 

where $a$ is the arithmetic mean^{}, $g$ the geometric mean^{} and $h$ the harmonic mean^{}. It is easy to see that

$$\frac{c+h}{2}=a\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}\sqrt{ah}=g.$$ 

Example. The integer 5 is the contraharmonic mean of 2 and 6, as well as of 3 and 6, i.e.
2, 5, 6, are in contraharmonic proportion, similarly are 3, 5, 6:

$$\frac{{2}^{2}+{6}^{2}}{2+6}=\frac{40}{8}=\mathrm{\hspace{0.33em}5},\frac{{3}^{2}+{6}^{2}}{3+6}=\frac{45}{9}=\mathrm{\hspace{0.33em}5}$$ 

Note 1. The graph of (2) is a quadratic cone surface ${x}^{2}+{y}^{2}xzyz=0$, as one may infer of its level curves^{}

$${\left(x\frac{c}{2}\right)}^{2}+{\left(y\frac{c}{2}\right)}^{2}=\frac{{c}^{2}}{2}$$ 

which are circles.
Note 2. Generalising (2) one defines the contraharmonic mean of several positive numbers:

$$c({x}_{1},\mathrm{\dots},{x}_{n}):=\frac{{x}_{1}^{2}+\mathrm{\dots}+{x}_{n}^{2}}{{x}_{1}+\mathrm{\dots}+{x}_{n}}$$ 

There is also a more general Lehmer mean^{}:

$${c}^{m}({x}_{1},\mathrm{\dots},{x}_{n}):=\frac{{x}_{1}^{m+1}+\mathrm{\dots}+{x}_{n}^{m+1}}{{x}_{1}^{m}+\mathrm{\dots}+{x}_{n}^{m}}$$ 

References

1
Diderot & d’Alembert: Encyclopédie. Paris (1751–1777). (Electronic version http://portail.atilf.fr/encyclopedie/here).

2
Horst Hischer: “http://hischer.de/uds/forsch/preprints/hischer/Preprint98.pdfViertausend Jahre Mittelwertbildung”. — mathematica didactica 25 (2002). See also http://www.math.unisb.de/PREPRINTS/preprint126.pdfthis.

3
J. Pahikkala: “On contraharmonic mean and Pythagorean triples^{}”. – Elemente der Mathematik 65:2 (2010).
Title 
contraharmonic proportion 
Canonical name 
ContraharmonicProportion 
Date of creation 
20150906 19:48:08 
Last modified on 
20150906 19:48:08 
Owner 
pahio (2872) 
Last modified by 
pahio (2872) 
Numerical id 
24 
Author 
pahio (2872) 
Entry type 
Definition 
Classification 
msc 26E60 
Classification 
msc 1100 
Classification 
msc 01A17 
Classification 
msc 01A20 
Related topic 
ProportionEquation 
Related topic 
Mean3 
Related topic 
PythagoreanHypotenusesAsContraharmonicMeans 
Related topic 
HarmonicMean 
Related topic 
ConstructionOfContraharmonicMeanOfTwoSegments 
Related topic 
ContrageometricProportion 
Defines 
contraharmonic mean 
Defines 
antiharmonic mean 