Dotty Circles

Today I was working with some teachers exploring dotty circles as an introduction into properties of cyclic quadrilaterals. We started by creating triangles using the centre dot and we found their angles. We then drew quadrilaterals that only used edge dots and used our previous angles to help us explore the angles in the shapes we had found.

It is great practice for finding missing angles, there are lovely patterns to explore and it leads onto a really nice proof, no algebra required!

Problem: Cyclic Quadrilaterals
Printable sheet: Dotty Circles

While we were exploring this we were recording our results on the board and two of the teachers had come up with the same angles though they had drawn different quadrilaterals. It led on to a fabulous discussion as to how many other ways these angles could occur, when this would happen and why this was happening.

My Questions: 
What quadrilaterals is it possible to make using the dots on a circle? 
Does it matter how many dots there are?