# Tag Archives: pythagoras

## Pythagoras, triples, 3,4,5, a calculator.

How to generate Pythagorean triples (example: 3,4,5), well one way at least.

Starting with (x + y)^{2} = x^{2} + y^{2} + 2xy and (x – y)^{2} = x^{2} + y^{2} – 2xy

we can write the difference of two squares

(x + y)^{2} – (x – y)^{2} = 4xy

and if we write x = A^{2} and y = B^{2} the right hand side is a square as well.

Thus:

(A^{2} + B^{2})^{ 2} – (A^{2} – B^{2})^{ 2} = 4A^{2} B^{2} = (2AB)^{ 2}

which can be written as

(A^{2} + B^{2})^{ 2} = (A^{2} – B^{2})^{ 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.

Have fun…….

My next post will be about finding the radius of the inscribed circle in a right angled triangle…..

## And now Pythagoras again, with bonus

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)

Filed under algebra, geometry, Uncategorized

## Pythagoras converse, proof from scratch

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: