Hello decor

Welcome to Math Help Forum!We have a triangle here, with sides of length units.

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:

and

The next gives:

and

... and so on.

Grandad