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In its initial position, the vertex has coordinates , where . If the triangle is rotated through an angle anticlockwise about , then this vertex will have coordinates .
The vertex initially at will now be at .
In order for these to be integer coordinates, then, all the following will need to be integers:
Pythagorean Triples would give a general method for solving this type of problem, but with the numbers and , the only possible values are based on , since no other triple has a largest number which is a factor of and (apart, obviously, from derivatives of this triple: , etc).
So I reckon that the possible values of are etc.
The first of these rotations gives coordinates:
The next gives:
... and so on.