13. Given a parallelogram with a fixed base and a side of fixed length, find the locus of the center of the parallelogram.
13. Establish a coordinate system as shown in the figure

Let's set $A(0,0), B(b, 0), |A D|=1$
Let $D$ be $\cos \theta + i \sin \theta$, and denote the center of the parallelogram as $O$
So, $z_{O} = \frac{1}{2} (z_{B} + z_{D}) = \frac{b + \cos \theta}{2} + \frac{i \sin \theta}{2}$.
Let $\frac{b + \cos \theta}{2} = x, \frac{\sin \theta}{2} = y$.
Therefore, $(2 x - b)^{2} + 4 y^{2} = 1$
Thus, the required locus is a circle.