The orthocenter is the point where the three altitudes intersect. There is a theorem which states that the three altitudes of a triangle will indeed have a point in common.
To find this common point, it suffices to develop equations for two of the altitudes and then solve for the point at which they intersect.
Working with the last two points, it can be seen that the altitude to the side defined by these two points is parallel to the x axis. This means that its slope is zero. Since the vertex (-2,1 ) is on the altitude line, the equation of this altitude is the horizontal line y = 1.
Working with the first and third points, it can be seen that the altitude to the side defined by these points has
slope = 1. (Because the altitude is perpendicular to the line joining these two points.)
So (y - y2) = 1 ( x - x2) is the equation of this altitude, with x2 = 3, y2 = 4. After rearrangement this is
y = x + 1.
Solving y = 1 and y = x + 1 together gives y = 1, x = 0.
So the coordinates of the orthocenter are (0,1)
