# Find Center of Circle given Radius, Circumference Point, and that Point's Rotation

I need to find the center point of a circle $(x,y)$ given:

• The radius $\mathbf r$ of the circle

• A point on the circumference of the circle $\mathbf (a,b)$

• The clockwise degrees of rotation $\mathbf t$ of the point $(a,b)$ about the center point $(x,y)$

Here's an illustration :

I've tried

$\begin{cases} x=a+r\;\cos(t) \\ y=b+r\;\sin(t) \end{cases}$

and it seemed to get me close but it's possible I'm missing an additional piece of the puzzle.

• you have inversed things, this is $a=x+r\cos(t)$ and $b=y-r\sin(t)$. – zwim Mar 14 '17 at 16:41

Consider that you know how to write $(a,b)$ using $x,y,\mathbf{r},\mathbf{t}$: $$\left\{ \begin{array}{l} a=x+\mathbf{r}\cos(2\pi-\mathbf{t}) \\ b=y+\mathbf{r}\sin(2\pi-\mathbf{t}) \end{array} \right.$$