What's the Point?

Here is a math puzzle that is as delightful as it is challenging. From the original problem that was posted on Usenet in April 2004, it has branched into a plethora of delightful variations, each with its own twists and turns. The search for solutions will take you on a wonderful journey through the lands of Geometry, Algebra, and Number Theory.

Here is the original problem, as stated:

Place a point P in a square so that the line segments from P to the

vertices have integer lengths, and no two have the same length.
If this is possible, what is the smallest such square?

And here are some interesting variations:

1) The same problem using a rectangle instead of a square. Smallest rectangle here meaning the one with the smallest area.

2) Finding a point on the midline of a square (the segment connecting the midpoints of opposite sides), where you need only two different integer lengths.

3) Finding a point on the diagonal of a square, where you need only three different integer lengths.

4) The same problem using an equilateral triangle instead of a square.

5) The same problem using a regular pentagon instead of a square.

6) The same problem using a regular hexagon instead of a square.

7) The same problem using a regular octagon instead of a square.

8) The same problem in three dimensions using a cube instead of a square.