The formula you give requires three points to determine the center, otherwise it is indeterminate (and could be any of infinitely many circles).
If you're just wanting to know whether it is clockwise or anticlockwise, suppose you have three points on the path, $p_1$, $p_2$, and $p_3$, and consider vectors $v_1=p_2-p_1$ and $v_2=p_3-p_2$. Let $R$ be the rotation matrix for the angle 90 degrees anticlockwise. Compute the number $a=(Rv_1)\cdot v_2$, where the dot is dot product. If this number $a$ is positive, the points are along an anticlockwise path, and if it is negative, the points are along a clockwise path. This formula represents projecting the second vector onto a vector perpendicular to the first, then seeing in which direction that projection goes.
Just to be more concrete, suppose $v_1=(x_1,x_2)$ and $v_2=(y_1,y_2)$. Then $a=x_1y_2-x_2y_1$. (This is the third component of a cross product of $v_1$ and $v_2$ as if they were 3D vectors.)