The orthocenter is the point where all three altitudes of the triangle intersect. 


Notice that the side of the triangle, connecting the vertices (-2,-1) and (6,-1), is horizontal line parallel to x-axis.

Therefore, the altitude drawn to this side is vertical line x = const, and since it passes through the point (2,5), this constant is equal to 2,
and the equation of this altitude is x = 2.


We will find the orthocenter as the intersection point of this altitude with the other altitude drawn from the vertex (6,-1).

This altitude is perpendicular to the side of the triangle connecting two other points, (-2,-1) and (2,5).

The slope of this side/segment is m =  =  =  = .


Since the altitude from the point (6,-1) is perpendicular to this side, it has the slope .
And since it passes through the point (6,-1), its equation is

y - (-1) = ,   or    

y + 1 = ,   or   y = .


Its intersection with the line x = 2 has y-coordinate  y =  =  .


Thus the orthocenter is the point (x,y) = (,).