questions about a certain elliptic curve and its points

[American Citizen <website.reader3@gmail.com>]

Hello:

I have a certain elliptic curve (1)

(1)  E(n) = [0, 0, 0, n^2, 0]

let's say that we pick any 3 points on this curve, P1[x1,y1], P2[x2,y2] 
and P2[x3,y3]

I am looking for these 3 points to satisfy the relationship in (2) below:

(2)  n^2 = x1*x2 + x1*x3 + x2*x3

where n is positive integer, and x1,x2,x3 are rational or integer.

How would one go about finding n, from arbitrarily picking x1, x2, and 
x3, if this is even possible?

I can pick n, and then subsequently find x1, x2 and x3; but most of the 
time the 3 points don't satisfy  (2)

I know that we can pick an arbitrary x1,x2, but then we must carefully 
pick x3 to satisfy the n^2 relationship, making n a positive integer, (I 
suspect that we could find more than one n value for a given x1,x2 pair, 
but how do we guarantee that the 3 points are on the [0,0,0,n^2,0] 
curve? This is fitting 3 points to an elliptic curve (usually this takes 
5 points, but they must be on an unknown elliptic curve to start with) 
so there is latitude on finding the other 2 points.

Any ideas? I know that making the ordinates from the points rational is 
part of the solution here.

Randall