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If value of $a+b+c$ is given as $n$ and it is provided that a2+b2 = c2. So, I need to find the value of $(abc)$ in terms of $n$.

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closed as off-topic by Dietrich Burde, user91500, José Carlos Santos, Shaun, Antonios-Alexandros Robotis Aug 9 '17 at 10:10

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You can't hope for an answer in general, because two equations in three variables will generally give a one dimensional set of points. But the answer can sometimes be given in terms of a single parameter. Here it is easy enough to work through:

$$a+b+c=n$$

$$a^2+b^2+c^2+2c(a+b)+2ab=n^2$$

$$2c^2+2c(n-c)+2ab=n^2$$

$$2cn+2ab=n^2$$

$$2c^2n+2abc=cn^2$$

$$abc=\frac{cn(n-2c)}2$$

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    $\begingroup$ Note that $2c\lt n$ is simply the triangle inequality. $\endgroup$ – Mark Bennet Aug 9 '17 at 9:00
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    $\begingroup$ I think the last line should be abc = cn(n - 2c)/2. $\endgroup$ – Abhinav Kushagra Aug 9 '17 at 9:29
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the points $(x,y,z)$ with $x^2+y^2=z^2$ form an infinite cone in the $x,y,z$ plane.

The points $(x,y,z)$ with $x+y+z=\alpha$ form a plane. The intersection of the plane and the cone is a conic.

This conic can contain a large number of lattice points.

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