completing square for a circle In the following question: 

I don't understand how we can get from the original equation to the final equation using completing the square.
Any thoughts as how to get to the final equation?

One way of doing this is to avoid fractions completely until the last step.
$$\begin{align}
\frac{x+y}{x^2+y^2+1}&=c\\
x+y&=c(x^2+y^2+1)\\
x+y&=cx^2+cy^2+c\\
cx^2-x+cy^2-y+c&=0\\
cx^2-x+cy^2-y&=-c\\
4c(cx^2-x+cy^2-y)&=4c(-c)\\
4c^2x^2-4cx+4c^2y^2-4cy&=-4c^2\\
(2cx)^2-2(2cx)+(2cy)^2-2(2cy)&=-4c^2\\
(2cx)^2-2(2cx)+1+(2cy)^2-2(2cy)+1&=2-4c^2\\
(2cx-1)^2 + (2cy-1)^2 &= 2-4c^2\\
\frac{(2cx-1)^2 + (2cy-1)^2}{4c^2} &= \frac{2-4c^2}{4c^2}\\
\bigg(x-\frac{1}{2c}\bigg)^2 + \bigg(y-\frac{1}{2c}\bigg)^2 &= \frac{1}{2c^2}-1\\
\end{align}$$
