Tag Archives: pythagoras
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…..
I was attempting to solve a geometrical problem the other day, a problem which, due to my complete misunderstanding, had no solution, when this popped out. It is probably bog-standard, but new to me, and this time I don’t have the heart to check if it is one of the 100 proofs of The Pythagoras theorem.
Now let theta be the angle ACB. Angle ABD is then 2*theta.
Set r = 1, then a is sin(2*theta) and b is cos(2*theta), and so
sin(theta) = (1 – b)/a = 1/a -b/a = cosec(2*theta) – cot(2*theta)
and cosec(theta) = (1 + b)/a = 1/a + b/a = cosec(2*theta) + cot(2*theta)
I’ve done most of the work,
so now you can show that sin(2*theta) = 2*sin(theta)*cos(theta)
There is a website with 100 proofs of the famous theorem of Pythagoras, but when I trawled the net looking for a proof of the converse, they all assume the basic theorem.
Here’s how to do it from scratch, which is considerably more satisfying, and also a simple application of similar triangles and basic algebra: