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In the image above which i horribly draw i am showing first a circle. The following "shapes" have small circles which represent points. All those points are positioned on the perimeter of that circle. Each and every point is directly connected with a straight line with all the other points.
My questions are:

A) Is there an equation / formula to find how many triangles appear given the number of points and circle radius?

B) Is there an equation / formula to find how many Equilateral triangles appear given the number of points and circle radius?

C) Is there an equation / formula to find how many Isosceles triangles appear given the number of points and circle radius?

D) Is there an equation / formula to find how many Scalene triangles appear given the number of points and circle radius?

I hope you understand my questions and sorry for that image. I have searched online but got nothing. Bare in mind that i have elementary math knowledge and am not so "bright" :D. Thank you in advance!

uhoh
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    Hello and welcome to math.stackexchange. You have drawn what is known as complete graphs with $n$ vertices, and your first question is about the number of complete subgraphs or cliques with three vertices. You may find it useful to learn more about these terms. As to your first question, the number is $\binom{n}{3}$, i.e. 4 for $n = 4$ vertices, 10 for $n = 5$ vertices, and so on. Simply pick any three vertices, and they will form a triangle. The radius of the circle is irrelevant. – Hans Engler Dec 22 '21 at 21:07
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    @HansEngler As the OP is a new contributor, he may be not familiar with the notation you used (even though it's pretty standard in combinatorics), so IMVHO it may be helpful to add a key word in plain English: binomial coefficient aka number of combinations. – CiaPan Dec 22 '21 at 21:15
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    Thank you both! I will study what you mentioned as soon as possible. –  Dec 22 '21 at 21:31
  • As I understand it, your n points are situated 1) on a circle 2) at the same distance one from each of its neighbours, i.e. are the vertices of what is called a regular polygon (same side lengths) for example a regular (convex or not) octogon. Right or not ? – Jean Marie Dec 22 '21 at 21:52
  • The n points are situated on the perimeter of the circle and the distance between them is not same. The point are , let's say, randomly placed. –  Dec 23 '21 at 04:36
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    If they are randomly placed on the circle, the answer to questions B) and C) is (almost surely) 0 equilateral triangles, and 0 isosceles triangles. – Jean Marie Dec 23 '21 at 10:02
  • Hmm, you might be right. Is there a formula to find out for sure? I want to apply some movement to the points and see if any patterns emerge. I do not know why, but I have the feeling that prime numbers are somehow involved. Also, I think that it had to do with how a molecule behaves regarding "space" and "energy". It also might make me understand better entropy. There is no point in observing isolated events since even the slightest unit of anything can rapidly grow into a major contributor. Sorry if what I am saying does not make any sense. Please let me know if there is a formula. –  Dec 23 '21 at 12:07
  • I wan't aware of you comment because you hadn't written @my_pseudo. I doubt prime numbers are involved here. You could consider instead (maybe) that you have $N$ sites which are vertices of a regular polygon inscribed in the circle and that $n \le N$ sites are chosen for being the vertices of your final polygon... In this way, you could have for example isosceles triangles. – Jean Marie Dec 23 '21 at 23:58
  • @JeanMarie I will try that approach. Though, it is not my goal to necessarily have isosceles or equilateral triangles. I want to know what kind of triangles can exist in any given moment while those points occur. Are there any patterns in terms of kinds of triangles and their position within that frame? If only a single kind of triangle can exist, what are the margins between various triangle positions. What are the differences? How does the number of points affect the frame? More stuff that i want to check as well. I am more interested in "space" and "energy" at the moment but i check all. –  Dec 24 '21 at 11:22
  • @JeanMarie I think triangle is the most interesting shape of all. It is so simple and beautiful. –  Dec 24 '21 at 11:24
  • If only the "topology" is involved, joining all the $n$ points in all the possible manners gives you a complete graph $K_n$. The number of triangles you can "extract" from this graph is $\binom{n}{3}=n(n-1)(n-2)/6$. Otherwise, if "metric" considerations come into the play (if it matters that a point is closer or farther from its neighbours on the circle), you have to consider an expression of the "energy" for example by summing all mutual squared distances of the different point pairs, and try to find if for example it has a connection with the individual energies of the triangles,.. – Jean Marie Dec 24 '21 at 12:06
  • @JeanMarie thank you very much. I will try what you mentioned as soon as possible. Merry Christmas! –  Dec 24 '21 at 13:07
  • @JeanMarie Hey, it has been baffling me what you said about prime numbers not being involved and i have been thinking about it. Apart from cryptography, i can't think of any other usage of prime numbers. Unless we think of their properties as algorithmic steps / constants. I am not sure how to even start testing that but that is what i am stuck at regarding prime numbers. That also is, the only way i can think of that prime numbers could be involved. What do you think, could prime number's properties be considered as a constant ? –  Dec 25 '21 at 00:28
  • @JeanMarie On this case for example: no matter what kind of triangles there can be, there can never be triangles forming a full circle. That's kind of stupid but you get the meaning. –  Dec 25 '21 at 00:40
  • I hope you don't mind that I made your graphic visible; interesting question! – uhoh Jan 25 '22 at 23:46
  • @HansEngler $\binom n3$ only counts those triangles whose vertices lie on the circle. It's not clear but I don't think that's what the OP intends. – bof Jan 26 '22 at 00:20
  • If $n$ points are placed "randomly" on a circle (so that no three chords are concurrent at a point inside the circle) then the number of triangles formed by the chords is $\binom n3+4\binom n4+5\binom n5+\binom n6.$ – bof Jan 26 '22 at 00:28
  • @Uhoh in the contrary, thank you! –  Jan 26 '22 at 03:56

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