Definition: The orthocenter of a triangle is the point where its three altitudes meet; the altitudes are the perpendicular segments from each vertex to the opposite side's line.
Where does the orthocenter lie?
Inside an acute triangle, at the right-angle vertex of a right triangle, and outside an obtuse triangle. For an obtuse triangle, the altitudes must be extended as lines to meet.
How do you find it with coordinates?
Write the equations of two altitudes and solve them as a system.
Example. Triangle with A(0, 0), B(8, 0), C(2, 6). The altitude from C to AB is the vertical line x = 2. Side BC has slope (6 – 0)/(2 – 8) = -1, so the altitude from A has slope 1: y = x. They meet where x = 2, so y = 2. The orthocenter is (2, 2). Checking with the third altitude: side AC has slope 3, so the altitude from B has slope -1/3, and y = -(1/3)(x – 8). At x = 2 this gives y = 2, which matches.
What mistakes occur?
Using the slope of the side instead of its negative reciprocal. Also, an altitude does not usually bisect the side, so do not use midpoints. The orthocenter, centroid, and circumcenter lie on one line, called the Euler line, in any triangle.