## Regular polygons and lattice points.

Se say that a point in the plane is a lattice point if both its coordinates are integers. Prove that for we can’t find a regular polygon with egdes, its vertices being lattice points.

**Solution:**

Looking at Pick’s Formula, we see that two times the area of a polygon with vertices in lattice points is an integer. But if we take an equilateral triangle with side of length with (distance formula) then its area is which is not a rational number. Contradiction. As suggested in the comments, we need to take care of the case separately, but this is a simple consequence of .

Suppose now and denote our polygon . Then the angles of such a polygon are greater than . Therefore, if we construct such that . Then is a regular polygon with vertices in lattice points, which is strictly inside the initial polygon. If exists, then we can repeat this construction as many times we want, but this is a contradiction, since there are only a finite number of lattice points inside .

I don’t think this works for n = 6, because then Q_1 = Q_2 = … = Q_6. But then P_1P_2Q_1 would be an equilateral triangle, which provides the contradiction.

Yes, you are right. The case should be treated separately, but as you point out, it is a consequence of the case .