  There are four characteristic circles for each triangle on a plane. All for
are tangential to the three straight lines containing the triangles' three
sides. Three are exterior circles, the fourth is the in-circle. When the
triangle is Pythagorean, the four diameters are integers. Consider a
Pythagorean triangle with the property that one leglength is a perfect(or
integer)square, and with one of the four diameters also a integer square.Of the
eight resulting combinations, we prove that only six are possible or can occur.
We then completely parametrically describe the six families; each corresponding
to one of the six combinations.
