So I was digging up some old shit I used to mess with when I was in middle school and high school and found one of my autism projects I forgot about. I used to be obsessed with various obscure math topics, before I ended up a dropout shut in NEET. One such thing was collecting data on regular polygons with every vertex connected to each other with a diagonal, since I had fun drawing them in school with a ruler and making cool looking patterns in the shape.

All I could really think to collect and keep track of in a spreadsheet were the obvious:
>number of diagonals
>number of diagonal intersections
>number of regions within the shape
>number of each type of region formed

After counting them by hand, eventually I could see patterns emerging, specifically in regular polygons with an odd number of sides... so I focused on them. For odd n-gons, the formula for # regions and # of intersections were pretty basic 4th degree polynomials. The number of diagonals was even simpler to derive, merely being n(n-3)/2.

What was really interesting though was counting the number of triangles out of the total number of regions formed by the diagonals. After getting enough of a sample size, the pattern was pretty interesting. Every odd sided regular polygon had a number of triangle regions equal to a prime number, multiplied by the number of sides of that regular polygon. The specific prime number was also predictable, as it was always the first prime number between a^2 and (a+1)^2, otherwise known as the Legendre's conjecture prime number sequence (https://oeis.org/A007491). In my example's case, n = 2a + 3

You can now extrapolate the number of triangle regions formed in any odd sided shape with all vertices connected.

>37 sided regular polygon, n = 37, a = 17
Triangle # = 37 x [first prime betweeen 17^2 and 18^2] = 37 x 293 = 10841

>153 sided regular polygon, n = 153, a = 75
Triangle # = 153 x [first prime betweeen 75^2 and 76^2] = 153 x 5639 = 862767