If the circumcentre of the triangle formed by the points (0,0),(2,−1)  and (−1,3) is at 52,52 then the coordinates of  its orthcentre are

The coordinates of the centroid are G0+2−13,0−1+33 i.e. 13,23. Let α,β be the coordinates of orthocentre H. The coordinates of circumcentre  are O′5    52    2.Since G divides O' H the ratio 1 : 2.∴ α+53=13 and β+53=23⇒α=−4,β=−3