What is the smallest area of a triangle with corners at (0, 0) and (27, 14) and another point with integer coordinates?
zero, if the 3rd corner is at (54,28) or any other point on the same line.
Now, if you want to exclude the degenerate case above, you want to find the minimum distance between the line segment and another point with integer coordinates. Again, there is no minimum area, since the 3rd point can be arbitrarily close to the line, if it's far enough away.
Any point with integer coordinates in the solution set of
y > 14/27 x
will do the job. Then the altitude of the extremely long and skinny triangle will be, assuming the point is at (h,k)
|14h-27k|/√925
You can pick h and k to make 14h-27k as small as you want.
So, let's assume that you want the 3rd point C to be between the perpendiculars at the two points.
The points closes to the line segment AB are
(0,1), (1,1)
(2,2), (3,2)
...
(26,14)
The point where |14x-27y| is smallest will produce the minimum area.