# Tag Archives: pythagoras

## Radius of the inscribed circle of a right angled triangle

Filed under algebra, geometry

## 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 = 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.

Thus:

(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.

Have fun…….

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

Filed under algebra, geometry, teaching

## 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)