Is there any relation between (a+b+c) and (abc)?

Question

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 $n$.

Answer 1

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:

$$a+b+c=n$$

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

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

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

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

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

Answer 2

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.