New relations

New Reations with Pythagorean triples

A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule:

a² + b² = c²

And when we make a triangle with sides a, b and c it will always be a right angled triangle

List of the First Few

Here is a list of the first few Pythagorean Triples (not including "scaled up" versions mentioned below):

(3,4,5)	(5,12,13)	(7,24,25)	(8,15,17)	(9,40,41)
(11,60,61)	(12,35,37)	(13,84,85)	(15,112,113)	(16,63,65)
(17,144,145)	(19,180,181)	(20,21,29)	(20,99,101)	(21,220,221)
(23,264,265)	(24,143,145)	(25,312,313)	(27,364,365)	(28,45,53)
(28,195,197)	(29,420,421)	(31,480,481)	(32,255,257)	(33,56,65)
(3,544,545)	(35,612,613)	(36,77,85)	(36,323,325)	(37,684,685)

c is the longest side of the triangle, called the "hypotenuse"

a and b are the other two sides such that a< c ,b< c but a+b > c.  It is noted that one of the numbers a or b is 3 or mutiplles of 3. Let

 a+b = c + α, where α is also a positive integer. Squaring both sides, we have

 α² +2cα – 2ab = 0 Solving for α, we get α = - c + [c² + 2ab]^1/2. i.e, c² + 2ab must be a square number.

It is again noted that the product of ab i.e., twice the area of the right angled triangle is equal to a multiple (β) of the circumference (a+b+c) of the  right angled triangle such that β = α/2 .Since β is also a positive integer, α must be even.