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Swaathi from Brighton College Abu Dhabi sent in this excellent solution:

I started this problem by first identifying the different triangles within the 9-dot circle.

Let's figure out the red triangle's angles:

Since a circle has an interior angle of 360 degrees, we can divide 360 by 9 to give us one angle of the triangle. This is because the 9 dots are evenly spaced which means that all the red triangles are identical.

Now, let's look at the green triangle:

If we apply the same technique to the other triangles as well:

Now for the second part of the problem regarding quadrilaterals created using the points on the circumference of the circle.

I think that these methods can be applied to all the circles so in order to prove this theory I tested it with the 10 dot circle.

Extension:

Swaathi claims that opposite angles in any quadrilateral within a circle add up to 180°. Is this enough evidence to prove it?