What are the coordinates of the center of the circumscribed circle of a triangle with the 3 vertices known?

Question

So I have 3 points A(1,3), B(-2,1), C(-3,-1). What are the coordinates of the center of the circumscribed circle and what is the radius of the circle?

Answer 1

A middle point of $AC$ it's $D(-1,1)$ and $m_{AC}=\frac{-1-3}{-3-1}=1$.

Thus, an equation of $DO$ it's $y-1=-1(x+1)$ or $y=-x$.

Thus, $O(x,-x)$ and since $OB=OA$, we obtain: $$(x+2)^2+(-x-1)^2=(x-1)^2+(-x-3)^2,$$ which gives $x=2.5$, $O(2.5,-2.5)$ and $R=\sqrt{4.5^2+3.5^2}=\sqrt{32.5}$.

Done!

Answer 2

Let's denote your vertices by the vectors $\bf a$,$\bf b$, and $\bf c$, and the circumcenter by the vector $\bf p$.

We have

$$ {\bf (p-a)\cdot(p-a)}=R^2 $$

$$ {\bf (p-b)\cdot(p-b)}=R^2 $$

$$ {\bf (p-c)\cdot(p-c)}=R^2 $$

Subtracting the 2nd eq. from the 1st and the 3rd from the 2nd eq. gives

$$ {\bf p\cdot(a-b)}=\frac{1}{2}\Big({\bf a\cdot a -b\cdot b}\big)=f $$

and

$$ {\bf p\cdot(b-c)}=\frac{1}{2}\Big({\bf b\cdot b -c\cdot c}\big)=g $$

which gives you a set of simultaneous equations for the components $p_x$ and $p_y$ in terms of the known vertex coordinates.

I'll pretend to be a math prof and leave the rest as an exercise for the reader :-)