Pythagorean triangle

The side lengths of any right triangleMathworldPlanetmath satisfy the equation of the Pythagorean theoremMathworldPlanetmathPlanetmath, but if they are integers then the triangle is a Pythagorean triangleMathworldPlanetmath.

The side lengths are said to form a Pythagorean tripleMathworldPlanetmath.  They are always different integers, the smallest of them being at least 3.

Any Pythagorean triangle has the property that the hypotenuseMathworldPlanetmath is the contraharmonic mean

c=u2+v2u+v (1)

and one cathetusMathworldPlanetmath is the harmonic mean

h=2uvu+v (2)

of a certain pair of distinct positive integers u, v; the other cathetus is simply |u-v|.

If there is given the value of c as the length of the hypotenuse and a compatible value  h as the length of one cathetus, the pair of equations (1) and (2) does not determine the numbers u and v uniquely (cf. the Proposition 4 in the entry integer contraharmonic means).  For example, if  c=61  and  h=11, then the equations give for  (u,v)  either  (6, 66)  or  (55, 66).

As for the hypotenuse and (1), the proof is found in [1] and also in the PlanetMath article contraharmonic means and Pythagorean hypotenuses.  The contraharmonic and the harmonic mean of two integers are simultaneously integers (see this article (  The above claim concerning the catheti of the Pythagorean triangle is evident from the identity


If the catheti of a Pythagorean triangle are a and b, then the values of the parameters u and v determined by the equations (1) and (2) are

c+b±a2 (3)

as one instantly sees by substituting them into the equations.


  • 1 J. Pahikkala: “On contraharmonic mean and Pythagorean triples”.  – Elemente der Mathematik 65:2 (2010).
Title Pythagorean triangle
Canonical name PythagoreanTriangle
Date of creation 2013-11-23 11:53:13
Last modified on 2013-11-23 11:53:13
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 15
Author pahio (2872)
Entry type Result
Classification msc 11D09
Classification msc 51M05