
Re: What's the next triple ?
Posted:
Sep 1, 2010 7:36 AM


> A triangle with sides 10,17,9 has an area which is > equal to the square of 1/6 th of its perimeter. > Neglecting multiples, find the next largest area > triangle which has the same characteristics i.e. A = > (p/6)^2. > All values are integers and are large.
Let {x,y} be a rational point on the curve x y (x + y) = 9 (x y  1)
Then a = (9  x) r , b = (9  y) r, c = (x + y) r give the sides of a triangle satisfying
the given condition: area = (p/6)^2 where p=perimeter. r is any integer.
Ex 1. x = 4 , y = 1/2 a = 5 r , b = 17 r/2 , c = 9 r /2 Choose r = 2 to clear the denominators
Ex 2. x = 116964 / 226745 , y = 255025 / 51186
The question is how do we find rational points on the curve?

