First, the conclusion and proof come from http://oeis.org/A005732/a005732.pdf, here is a simple translation and processing
Conclusion#
The conclusion is that the number of triangles formed by connecting $n$ points on a circle is $C_{n+3}^6+C_{n+1}^5+C_n^5$
Proof#
How do we prove this? Let's start by considering how many triangles can be formed from any number of points among the $n$ points on the circle. We first look at the diagram with 7 points to observe! 
If you have plenty of patience, you can count that there are $287$ triangles.
However, we want to derive a general formula. We can observe that each triangle is composed of multiple or $0$ points on the circle, so let's categorize this discussion.
Classifying Cases#
First, a tip: recognize the concept of "on the circle": it means on the edge of the circle.
- The number of triangles formed by $3$ points on the circle is $C_n^3$. Proof: Any three points on the circle can form a triangle, and vice versa. (No need to draw a diagram for this, right? qwq)
- The number of triangles formed only by $2$ points on the circle and $1$ other point is $4 \times C_n^4$. Proof: For every $4$ points on the circle, $4$ triangles can be formed "with $2$ points on the circle"
- The number of triangles formed only by $1$ point on the circle and $2$ other points is $5 \times C_n^5$. Proof: For every $5$ points on the circle, $5$ triangles can be formed "with $1$ point on the circle"
- The number of triangles formed only by $0$ points on the circle and $3$ other points is $C_n^6$. Proof: For every $6$ points on the circle, $1$ triangle can be formed "with $0$ points on the circle"
(Of course, this diagram also includes connections between other points, but I was too lazy to draw them emmmm)
Organizing Cases#
Finally, we need to organize this expression (it's quite complex qwq) (reference this)