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

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

Mathematics
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Draw it :)
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 purpo|dw:1327108062011:dw|ses).

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