A triangle has vertices at (1, 2), (3, −4) and (5, −6). What are the coordinates of its circumcentre?

The circumcentre $(x,y)$ is equidistant from all three vertices. Equating the distances to (1, 2) and (3, −4) gives $x-3y=5$; equating those to (3, −4) and (5, −6) gives $x-y=9$. Solving, $y=2$ and $x=11$. Check: the distance from (11, 2) to each vertex is 10. The point (3, −8/3) is the centroid, not the circumcentre.