triple when they fail the test. For example, the triples and each pass the test that is a perfect square, but neither is a Pythagorean triple. When a triple of numbers , and forms a primitive Pythagorean triple, then and one-half of are both perfect squares; however this is not a sufficient condition, as the numbers pass the perfect squares test but are not a Pythagorean triple since . At most one of , , is a square. The area of a Pythagorean triangle cannot be the square or twice the square of a natural number. Exactly one of , is divisible by 2 (is even), but never . Exactly one of , is divisible by 3, but never . Exactly one of , is divisible by 4, but never (because is never even). Exactly one of , , is divisible by 5. The largest number that always divides abc is 60. Any odd number of the form , where is an integer and , can be the odd leg of a primitive Pythagorean triple [PPT]. See almost-isosceles PPT section below. However, only even numbers divisible by 4 can be the even leg of a PPT. This is because Euclid's formula for the even leg given above is and one of or must be even. The hypotenuse is the sum of two squares. This requires all of its prime factors to be primes of the form . Therefore, c is of the form . A sequence of possible hypotenuse numbers for a PPT can be found at . The area is a congruent number divisible by 6. In every Pythagorean triangle, the radius of the incircle and the radii of the three excircles are natural numbers. Specifically, for a primitive triple the radius of the incircle is , and the radii of the excircles opposite the sides , 2mn, and the hypotenuse are respectively , , and . As for any right triangle, the converse of Thales' theorem says that the diameter of the circumcircle equals the hypotenuse; hence for primitive triples the circumdiameter is , and the circumradius is half of this and thus is rational but non-integer (since and have opposite parity). When the area of a Pythagorean triangle is multiplied by the curvatures of its incircle and 3 excircles, the result is four positive integers , respectively. Integers satisfy Descartes's Circle Equation. Equivalently, the radius of the