I came across a probability question which asks that when a triangle is formed by joining any 3 random vertices of a hexagon, what is the probability that the triangle will be an equilateral one. The correct answer to this problem in solution book is $$\frac{2}{^6C_3}$$ I was wondering if there is a proper method to formulate the total number of equilateral triangles which can be formed by joining the vertices of a polygon with N sides. I asked the same question in a discord group and someone gave me this relation without a proof $$\text{the number of equilateral triangles by joining vertices * 3 = the number of sides in polygon}$$
If this relation is true (it works for the above probability question), is there any way we can prove it?