andrenoble wrote:
It's a very good question, I would like to add my approach:
The area of parallelogram is a * h, where a is base, and h is height.
(1) All coordinates are integers
Since we know that all coordinates are integers, it means that a will be an integer (e.g., x4 - x1 will be an integer), and h will be an integer as well. It gives us multiplication of two integers, and hence - an integer.
(2) is insufficient, since all sides could be equal to a non-integer, which will give a non-integer value. At the same time, we could have a figure will all integer coordinates and equal sides.
Hope it helps someone.
For statement 1, the case that you've considered holds only if one of the sides is aligned to one of the axes, i.e. two of the sides are || to x-axis or two of the sides are || to y-axis.
If none of the sides are || to any of the axes, the dimensions of base and height would have to be calculated using distance formula.
Now even if we use integral values in the distance formula, we may get roots, i.e. non-integral values. So how do we eliminate this second case?