For questions on polygons, a flat shape consisting of straight lines that are joined to form a closed chain or circuit

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3
votes
2answers
1k views

Number of triangles in a regular polygon

A regular polygon with $n$ sides. Where $(n > 5)$. The number of triangles whose vertices are joining non-adjacent vertices of the polygon is?
7
votes
1answer
378 views

Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number

Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number, where a number $q$ is practical if and only if every integer less than or equal to ...
2
votes
2answers
64 views

Polygon and Pigeon Hole Principle Question

Seven vertices are chosen in each of two congruent regular 16-gons. Prove that these polygons can be placed one atop another in such a way that at least four chosen vertices of one polygon coincide ...
2
votes
2answers
281 views

Proving the regular n-gon maximizes area for fixed perimeter.

It is often assumed that, given $n$, the regular $n$-gon will make the most efficient use of perimeter for area. I have never seen this proven. Anyone have something slick? (That is, how can we ...
1
vote
1answer
53 views

Find self-avoiding-filling-polygon represented by system of linear equations

In Euclidean graph where each vertex is a point on the $2D$ plane, so the weight of each edge is the Euclidean distance between the vertices. I want to find self-avoiding-filling-polygon from my graph ...
1
vote
1answer
303 views

Relationship between the sides of inscribed polygons

In my math textbook there's a demonstration for the calculus of the circumference of a circle that involves regular polygons inscribed in the circle, but I don't get it. The book gives the following ...