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Topic: What's the next triple ?
Replies: 7   Last Post: Sep 1, 2010 7:36 AM

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 Ara M Jamboulian Posts: 71 Registered: 12/6/04
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?

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