We are asked to find the orthocenter of a triangle given the coordinates of the vertices:
First, we know that the orthocenter is the point of concurrency of the altitudes of the triangle. In the case of a right triangle, the vertex of the right angle is the orthocenter.
If the triangle is not right, we need only find two altitudes and their intersecion will be the orthocenter.
Given triangle ABC:
Find the slope of segment AB and segment AC. (Any two of the segments works.) Then find the slope of the lines perpendicular to the segments. (This will be the opposite reciprocal of the slopes calculated.
Now find the equation of the line through point C with the slope of the line perpendicular to AB. Simarlarly find the equation of the line through point B with the slope of the line perpendicular to AC.
The intersection of the two lines is the orthocenter.
Example: Given A(5,5),B(6,4) and C(0,4)
The slope of AB is -1 so the line perpendicular has slope 1.
The slope of AC is 1/5 so the slope of the line perpendicular is -5.
The equation of the line through C perpendicular to AB is y-4=1(x-0) or y=x+4
The equation of the line through B perpendicular to AC is y-4=-5(x-6) or y=-5x+34
x+4=-5x+34 ==> 6x=30 or x=5. Then y=9. The orthocenter is at (5,9)