How to generate Pythagorean triples (example: 3,4,5), well one way at least.
Starting with (x + y)2 = x2 + y2 + 2xy and (x – y)2 = x2 + y2 – 2xy
we can write the difference of two squares
(x + y)2 – (x – y)2 = 4xy
and if we write x = A2 and y = B2 the right hand side is a square as well.
(A2 + B2) 2 – (A2 – B2) 2 = 4A2 B2 = (2AB) 2
which can be written as
(A2 + B2) 2 = (A2 – B2) 2 + (2AB) 2
the Pythagoras form.
Now just put in some integers for A and B
2 and 1 gives 3,4,5
Conjecture1: This process generates ALL the Pythagorean triples.
Conjecture2: Every odd number belongs to some Pythagorean triple.
My next post will be about finding the radius of the inscribed circle in a right angled triangle…..