Let A = (5, -2), B= (-1, 2), C= (1, 4)

First write the equation of the line BC. Take any general point D on BC. Find the equation of the line AD.

Use (Slope of AD)* (Slope of BC) = -1.

Do the same thing for line AB or AC.

Find the intersection point of the lines AD and AE (where E lies on AB or AC whichever you choose).

That intersection point will be your required orthocentre.