Here come those Pythagoreans again! The special right triangles related to Pythagorean triples had been studied and used by the Babylonians and the Egyptians long before Pythagoras, but those are the vagaries of history....
Pythagoras did find a formula for an infinite number of these, so there's some justification for the name.
Pythagorean triple: a set of three integers x, y, z such that
. The triple is said to be primitive if gcd(x,y,z)=1.
Table: Examples of Pythagorean triples. Which are primitive?
Pythagorean triangle: a right triangle whose sides are of integral length.
Pythagorean theorem: a famous theorem about right triangles (not necessarily Pythagorean triangles), infamously misstated by the scarecrow in the Wizard of Oz http://mathworld.wolfram.com/PythagoreanTheorem.html : After receiving his brains from the wizard in the 1939 film 'The Wizard of Oz', the Scarecrow recites the following mangled (and incorrect) form of the Pythagorean theorem, ``The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.''
Lemma 1: If x, y, z is a primitive Pythagorean triple, then one of the integers x or y is even, while the other is odd.
Lemma 2: If 
, where gcd(a,b)=1, then a and b are 
powers. That is, there exist positive integers 
and 
for which
and
.
Theorem 11.1: All solutions of the Pythagorean equation
satisfying the conditions
are given by the formulas
For integers s>t>0 such that gcd(s,t)=1 and 
.
The radius of the inscribed circle of a Pythagorean triangle is always an integer (Theorem 11.2).