We are asked to find the circumcenter and orthocenter for triangle ABC with vertices at A(5,7),B(0,2) and C(1,1).
(1) The circumcenter is the intersection of the perpendicular bisectors of the sides of the triangle.
One way is to find the midpoints of the 3 sides which are (3,4), (5/2,9/2) and (1/2,3/2). Then we find the slopes of the sides, and find the equation of the line perpendicular to the side through the midpoint:
The slope of AC is 3/2, so the perpendicular bisector through the midpoint (3,4) has slope -2/3. The equation is y=-2/3x+6
The slope of AB is 1, so the perpendicular bisector through (5/2,9/2) has slope -1. The equation is y=-x+7.
The intersection of the lines is at (3,4).
** An easier way to get this is to realize that triangle ABC is a right triangle. (The slope of BC=-1 and the slope of BA=1 so the sides are perpendicular.) The circumcenter of a right triangle is the midpoint of the hypotenuse, in this case (3,4).
(2) The orthocenter is the intersection of the altitudes. Since the triangle is a right triangle, the altitudes meet at B (0,2) (B is the right angle.)