Put four distinct points anywhere on the plane. Connect each pair of these
points with line
segments. This yields six line segments (which might cross each other and
some might even
overlap, depending on the locations of the four points). The line segments
might have very
different lengths.
If the four points are located at the corners of a square, then these
segments will be the four
sides of the square and the two diagonals, which are longer than the sides.
Here are two problems to start you thinking:

Are there other four point configurations on the plane, other than squares,
which give rise to segments of only two different lengths?

How many essentially different four point configurations on the plane can
exhibit segments of
just two different lengths? What does essentially different mean?
