Squares from Diagonals

A net made from rope
If I start by drawing a diagonal, will I always be able to draw a square from it?
You might want to draw some squares of your own using the interactivity above, or on dotty paper like in the video below.


Here are some questions you might like to consider:

Given a diagonal, how can I quickly work out where the corners of the square will be?

When will all four corners of the square be on dots?