This paper is written on a level accessible to college/university students of mathematics who are taking second-year, algebra based, mathematics courses beyond calculus I. This article combines material from geometry, trigonometry, and number theory. This integration of various techniques is an excellent experience for the serious student. The author recalls that a Pythagorean triangle is a right triangle with integer side lengths, and that the hypotenuse of a Pythagorean triangle is also the diameter of its circumscribed circle. The radius R in a Pythagorean triangle will be an integer if the hypotenuse is an even number; otherwise it will be half an odd integer. The Pythagorean triples (6, 8, 10), (10, 24, 26) have R equal to 5 and 13, respectively, while the Pythagorean triples (9, 12, 15), (5, 12, 13), (21, 20, 29) have R equal to 15/2, 13/2, and 29/2, respectively. In particular, every primitive Pythagorean triangle has R equal to half an odd integer. On the other hand, in any Pythagorean triangle, the internal radius [rho] is, in fact, an integer. (Contains 2 figures.)