Other ways to determine a circle

In general, there is a whole family of circles passing through two given points $z_1$ and $z_2$. Some other ways to determine one circle from these two points are

• Giving the direction (or tangent vector) at one of the two points $z_1$ or $z_2$.
• Using a known symmetry of the circle.

As an example, suppose we have a circle that passes through $-1$ and $1$ on the real axis. Suppose we want the tangent vector to the circle at 1 to make angle $\pi/3$ with the real axis (counterclockwise). The center lies on the perpendicular bisector, which is the imaginary axis. Thus, the center is $t i$ for some real number $t$. Assume for convenience $t>0$. The radius connecting $t i$ to 1 makes angle $\arctan t$ with the line segment $[-1,1]$. To meet our goal, this angle must be $\pi/6$. That means $t=\tan (\pi/6)=1/\sqrt{3}$. Hence the center of the circle is $i/\sqrt3$ and the radius is $\sqrt{(1/\sqrt3)^2+1^2}=2/\sqrt3$.

Instead, we could have required that the circle be symmetric about the $x$-axis. That means $\bar{C}=C$. Then the center satisfies this condition, too, implying that the center is 0, from which we deduce the radius is 1.