#Let# #A = (3,1)#

#Let# #B = (1,6)#

#Let# #C = (2, 2)#

Equation for altitude through A:

#x(x_3-x_2)+y(y_3-y_2)=x_1(x_3-x_2)+y1(y_3-y_2)#

#=>x(2-1)+y(2-6)=(3)(2-1)+(1)(2-6)#

#=>x-4y=3-4#

#=>color(red)(x-4y+1=0)#-----(1)

Equation for altitude through B:

#x(x_1-x_3)+y(y_1-y_3)=x_2(x_1-x_3)+y2(y_1-y_3)#

#=>x(3-2)+y(1-2)=(1)(3-2)+(6)(1-2)#

#=>x-y=1-6#

#=>color(blue)(x-y+5=0#-----(2)

Equating (1) & (2):

#color(red)(x-y+5)=color(blue)(x-4y+1#

#=>-y+4=1-5#

#=>color(orange)(y=-4/3#-----(3)

Plugging (3) in (2):

#color(blue)(x-4)color(orange)((-4/3))color(blue)(+1)=0#

#=>color(violet)(x=-19/3#

The orthocenter is at #(-19/3,-4/3)# OR #(-6.333...,-1.333...)#

which is actually outside the #triangle# because the #triangle# is an obtuse #triangle#. Click here to find more.