Hjelmslev's Theorem

The applet bellow illustrates what's known as Hjelmslev's theorem, named after the notable Danish mathematician Johannes Hjelmslev (1873-1950):

If two line segments $I_{1}$ and $I_{2}$ are related by an isometry $f: P_{1} \rightarrow P_{2},$ then the midpoints between the corresponding points $P_{1},$ $P_{2}$ are either all different and collinear or all coincide.

A Java illustration and a more general statement have been posted on an earlier page. Here the illustration is written in JavaScript and is thus available on a greater number of devices.

Whatever looks like a circle in the illustration is draggable.