Rhombus Diagonals

I wonder what rhombuses I can draw if I start with a diagonal...

You might want to draw some rhombuses 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:

Can ANY two points form a diagonal of a rhombus?

Is there a way to work out where the vertices of the rhombus could be?

How many rhombuses can you draw for a given diagonal?