Puzzle answer

Ten circles.

One circle and only one circle can be drawn through any 3 dots.* Find the number of groups of 3 dots that are possible and you have found the number of possible circles. We will make a list of all the combinations and then count the entries in the list.

To help visualize the problem, draw five dots more or less evenly distributed in a circle. Number the dots 1 through 5. Consider three dots as  being a pair plus one. That might make it easier to see a pattern that can be repeated until all the possibilities have been gone through. The following list is produced by going through all the pair plus one combinations.

The pair 1 and 2, plus dot number 3.
The pair 1 and 2, plus dot number 4.
The pair 1 and 2, plus dot number 5.
The pair 1 and 3, plus dot number 4.
The pair 1 and 3, plus dot number 5.
The pair 1 and 4, plus dot number 5.
The pair 2 and 3, plus dot number 4.
The pair 2 and 3, plus dot number 5.
The pair 2 and 4, plus dot number 5.
The pair 3 and 4, plus dot number 5.

Since the list is 10 items long, the number of cirles possible is ten.

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*Well, almost any. If the 3 dots are in a straight line, then one circle cannot be drawn through all 3 dots.