Take any two vertices, write the equation of the line joining them. Next wite the equation of a line passing through the third vertex and perpendicular to the previous line.  This gives you the perpendicular from the third vertex

Rapeat the steps for another pair of vertices to get the perpendicular from another third vertex. You get the equation for another perpendicular from a different vertex.

Solve the equations of the perpendiculars to get the coordinates of the orthocenter