Finding coordinates of some points in picture

Question

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]

Answer 1

Hints:

• $O_1$ 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 $C$.