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How many triangles with area one can be constructed using these 7 points on the unit square lattice?

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I found 12 using simple counting but it may cause a lot of errors. The answer is true but I want a mathematical way to count them.

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    With what area? – 2017-02-28
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    @MichaelBurr $1$ – 2017-02-28
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    So each point is separated by $1$ unit right ? – 2017-02-28
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    To clarify: are those supposed to be the points $(0, 2), (0, 1), (1, 1), (2, 1), (0, 0), (1, 0), (2, 0)$? Or are they slightly perturbed from those locations, as the drawing might suggest? – 2017-02-28
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    This is a programming problem, rather than a mathematical one. I took the coordinates provided by John Hughes, made a set of all combinations of 3 points (there were $\binom73=35$ of those), and looked at the areas. There are 4 things with area $0$ (degenerate triangles, so to say), 16 triangles with area $1\over2$, 12 with area 1, one with area $3\over2$, and two with area $2$. – 2017-02-28

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First suppose that you do not use the top left vertex, then we want the triangles of maximum order which are $3+3$ ($3$ when the repeated vertices are in the bottom and another $3$ when the repeated vertices are on top).

Now lets count the triangles that use the vertex on the top left corner:

First count the triangles in which the other two vertices are in the same level, this gives us $3$ triangles.

After this we check the $3\times 3$ triangles in which the other two vertices are at different levels, this is a bit of casework but we can do it carefully and systematically.