TITLE: Rational solutions to $x^4+y^4=cz^2$
QUESTION [5 upvotes]: Suppose $c\neq 1$ is a squarefree number, and consider the curve $x^4+y^4=cz^2$. How can I find rational points on this curve?
What I really want to know is how to transform this into an elliptic curve. I'm confused for the following reason: this curve has three variables, while elliptic curves have two variables.

REPLY [3 votes]: From,
$$x^4+y^4=cz^2\tag1$$
If you do the following change of variables, 
$$X = -\frac{4 x^{2} y^{2}}{z^{2}}\tag2$$ 
$$Y = \frac{4 xy(x^{4} - y^{4})}{z^{3}}\tag3$$ 
then you can check that,
$$Y^2 = X^3 - 4c^2X\tag4$$ 
which is an elliptic curve in Weierstrass form, and is famously related to congruent numbers.