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Let $A(0,0)$, $B(2,0)$, $C(c_{x}, x_{y})$, $D(d_{x}, d_{y})$.
$O_1$ and $N$ is the center of circles (ABD) and (CKL).
Find coordinates of $C, N, K, L, O_1$.![Don't take care about coordinate system in picture - wrong numbers!][1]

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Typo: $B=(4,0)$. You can determine $O_1$ using the fact that it is equidistant of $A,B,D$. So it is the intersection of the 2 mediatrix, and of course, it will be in function of $d_x,d_y$. –  Sigur Sep 6 '12 at 22:34
There seems to be a lot of unstated information. For example, is $d_y=2$? Is this a parallelogram? –  Henry Sep 6 '12 at 22:50

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  • $O_1$ is on the perpendicular bisectors of $AB$ and $AD$, so for example its $x$-coordinate is $2$.

  • $N$ is twice as far away from $A$ as $O_1$, in the same direction.

  • If this is a parallelogram, the coordinates of $C$ are the sum of the coordinates of $B$ and $D$.

  • When calculating the coordinates of $L$ and $K$ you will have to solve quadratic equations, with two solutions, one of which gives $C$.

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