Question

If the vertices of triangle ABC are A(2, -1), B(0, 2) and C(4, -3), what are the coordinates of the circumcenter?

Answer 1

Draw it :)

Answer 2

It is a very obtuse triangle. The three points lie almost on a straight line.

Back to basics, a circumcentre is a point which is equidistant to all three points, so that it is the centre of a circle that passes through all three points.

Using Pythagoras theorem, and assuming the required circumcentre is at (x0,y0), then
R=radius of circumcircle
and
R^2=(Xa-X0)^2+(Ya-Y0)^2=(Xb-X0)^2+(Yb-y0)^2=(Xc-x0)^2+(Yc-y0)^2
You will have a system of two linear equations (since all the x0^2 and y0^2 terms cancel out) from which you can solve for X0 and Y0.

(Xa-X0)^2+(Ya-Y0)^2=(Xb-X0)^2+(Yb-y0)^2
reduces to 6y-4x+1=0
and
(Xb-X0)^2+(Yb-y0)^2=(Xc-x0)^2+(Yc-y0)^2
reduces to -10y+8x-21=0
from which you can solve for X0 and Y0.
Hint: X0+Y0=24 (for checking purposes).