The orthocenter of a triangle is the point of intersection of the three altitudes of the triangle. An altitude of a triangle is a line segment perpendicular to a side of the triangle with one end point on that side and the other endpoint at the opposite vertex of the triangle.
So your process needs to be:
Step 1: Select two of the vertices of the triangle. Using the slope formula and the coordinates of the two selected vertices, compute the slope of the line containing the line segment between the two vertices.
Use
where
Step 2: Use the slope you just calculated to determine the slope of a line perpendicular to the line containing the side of the triangle that you are working with. The slope of a perpendicular is the negative reciprocal, that is:
Step 3: Using the slope of the perpendicular you just calculated and the coordinates of the opposite vertex (that is, if you were working with side ST, you now use the coordinates of U), use the point-slope form of the equation of a line to write an equation for the line containing the altitude segment.
Use
Where
Repeat steps 1 through 3 using a different pair of points. Call the result Equation 2.
Equation 1 and 2 form a two-equation, two-variable system of equations, the solution set of which is the ordered pair containing the coordinates asked for in the original question. Solve the system.
If you have enjoyed yourself so far, you can repeat steps 1 through 3 for heretofore unused pair of points to create Equation 3. Then use either Equation 1 or Equation 2 along with Equation 3 to create a second system of linear equations. Presuming correct work throughout, the solution to this new system must be identical to the first solution you obtained -- giving you a method to check your work.
John