incircle radius determined by Pythagorean triple

If the sides of a right triangleMathworldPlanetmath are integers, then so is the radius of the incircleMathworldPlanetmath of this triangle.
For example, the incircle radius of the Egyptian triangle is 1.

Proof.  The sides of such a right triangle may be expressed by the integer parametres m,n with  m>n>0  as

a= 2mn,b=m2-n2,c=m2+n2; (1)

the radius of the incircle ( is

r=2Aa+b+c, (2)

where A is the area of the triangle.  Using (1) and (2) we obtain


which is a positive integer.

Remark.  The corresponding radius of the circumcircleMathworldPlanetmath need not to be integer, since by Thales’ theorem, the radius is always half of the hypotenuseMathworldPlanetmath which may be odd (e.g. 5).

Title incircle radius determined by Pythagorean triple
Canonical name IncircleRadiusDeterminedByPythagoreanTriple
Date of creation 2013-03-22 17:45:46
Last modified on 2013-03-22 17:45:46
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Feature
Classification msc 11A05
Synonym incircle radius of right triangle
Related topic Triangle
Related topic PythagoreanTriple
Related topic DifferenceOfSquares
Related topic FirstPrimitivePythagoreanTriplets
Related topic X4Y4z2HasNoSolutionsInPositiveIntegers