incircle radius determined by Pythagorean triple
If the sides of a right triangle are integers, then so is the radius of the incircle
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 with as
(1) |
the radius of the incircle (http://planetmath.org/Incircle) is
(2) |
where is the area of the triangle. Using (1) and (2) we obtain
which is a positive integer.
Remark. The corresponding radius of the circumcircle need not to be integer, since by Thales’ theorem, the radius is always half of the hypotenuse
which may be odd (e.g. 5).