Suppose a circle in a 2 dimensional plane, given two points on the circle, i.e. $(x, y)^T, (x', y')^T$ and the angle between the tangle line at $(x, y)^T$ and horizontal axis is $\theta$. We have the coordinates of the center of the circle as follows
$\begin{pmatrix}x_c \\ y_c\end{pmatrix} = \begin{pmatrix}\frac{x + x'}{2} + \frac{1}{2}\frac{(x-x')\cos(\theta)+(y-y')\sin(\theta)}{(y-y')\cos(\theta)-(x-x')\sin(\theta)}(y - y') \\ \frac{y + y'}{2} + \frac{1}{2}\frac{(x-x')\cos(\theta)+(y-y')\sin(\theta)}{(y-y')\cos(\theta)-(x-x')\sin(\theta)}(x' - x)\end{pmatrix}$
How to derive this formula ? The only hint that "this results from the center of circle lies on a ray that lie on the half-way point between $(x, y)^T$ and $(x', y')^T$ and is orthogonal to the line between these coordinates. "