We draw the triangle:



The orthocenter is where all three of the extended altitudes intersect.

We draw the altitude from B to AC





Since it's vertical and goes through B(12,6) its equation is 

x = 12

Now we draw another altitude from A to BC. but to do that we have to first
extend BC.





To find the equation of this second (red) altitude, we need to know its slope.
It is perpendicular to BC.  So first we find the slope of BC:
A(0,0) B(12,6) C(18,0)



So the slope of the red altitude is the opposite-signed reciprocal of
the slope of BC.  The opposite signed reciprocal of -1 is .

Since it goes through the origin, its y-intercept is b=0 and so its
equation is 

y = 1x + 0 

or

y = x

To show where the green altitude intersects the red altitude,
we must extend both of them:


  
We just need to find the point where the red altitude and the 
green altitude intersect, for that is the orthocenter.

We solve the easy system:



That has the solution (12,12),

so the orthocenter is O(12,12)



We didn't need the third altitude. If we were to draw it we 
would find that it would also go through the orthocenter:

Edwin