A triangle has vertices at (0, 0), (6, 0) and (0, 8). What are the coordinates of its circumcentre?

The triangle is right-angled at the origin, since the legs lie along the axes. In a right triangle the circumcentre is the midpoint of the hypotenuse, so it is the midpoint of (6, 0) and (0, 8), which is (3, 4). Note (2, 8/3) is the centroid, not the circumcentre.