Given a fixed distance $2a$, and two points $(F_1,F_2)$ in the Euclidean plane, one can define an ellipse as the set of points $E$ such that the sum of the distances $d(E,F_1) + d(E,F_2)$ is equal to $2a$.
How can one prove that those two foci $F_1$ and $F_2$ are unique or not? In other words, prove that for any other set of points $(F_1',F_2') \neq (F_1,F_2)$, and some distance $2a'$, there exists at least one point $P$ in the previously defined ellipse $E$, such that $d(P,F_1') + d(P,F_2') \neq 2a'$. It would be nice if there was a nice way of finding one or even all points P with that property. That means I wish the solution to be as constructive as possible, even though I will not discard a proof by contradiction.
A weak proof can be made using the Euclidean distance formula $d(A,B)=\sqrt{(x_a - x_b)^2 + (y_a - y_b)^2}$. A strong proof would require using a general distance formula using these properties.