The right triangle with sides of length 3, 4, and 5 can be drawn in the coordinate plane so that it's vertices have integer coordinates (for example, (0,0), (4,0), (0,3)).

- Can you find another example of a triangle in the plane whose vertices have integer coordinates and whose sides have lengths that are consecutive integers?
- Can you find an infinite family of such triangles?
- Can you characterize such triangles?