TITLE: System of equations: $x^2+y=7, y^2+x=11$
QUESTION [5 upvotes]: Possible Duplicate:
Steps to solve  this system of equations 

During the flight from Moscow to Yerevan my neighbor gave me the following problem:
Solve the system:
$$\left\{\begin{array}{c}x^2+y=7 \\ y^2+x=11. \end{array}\right.$$
It is easy to find 1 of the 4 solutions. Is there a beautiful way to find the other three?

REPLY [6 votes]: This is maybe a half solution, maybe less. I tried several things and maybe these ideas inspire someone to write a complete solution. 
Note that the intersection points lie on the circle of the form
$$\left(x+\frac12\right)^2+\left(y+\frac12\right)^2=\frac{37}{2}$$
This follows from a general theorem on the intersections of two parabolas with orthogonal axes of symmetry.
Now there are so many theorems on cyclic quadrilaterals and even some statements on a parabola passing through the four corners. Then the given point $(2,3)$ should give some additional information. But I couldn't find a concise way of finishing this.