Considering the/a definition of a regular polygon from Wiki : In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). , my question is how to prove that the number of symmetry lines is equal to the number of […]

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 formula: $$l_{2n}=\sqrt{2R^2 – R\sqrt{4R^2-l_{n}^2}}$$ Where $l_{n}$ is the side of a regular n-sided polygon inscribed in a circle with R radius. […]

In the definition of regular polygon at the Wikipedia, there is this statement about the limit of a n-gon: “In the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed.” My question is: I […]

How many ways are there to create a quadrilateral by joining vertices of a $n$- sided regular polygon with no common side to that polygon? It’s quite easy to solve for triangles for the same question, logic remains same, we need to choose $4$ vertices with none of them being consecutive, what I did is […]

I am trying to find the vertices of a regular polygon using just the number of sides and 2 vertices. After the second vertex, I will make left turns to find each subsequent vertex that follows. For example, If I have 4 sides, and 2 points, (0,0) & (0,10), how would I go about find […]

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 with some of the chosen vertices of the other one. If possible, please provide a detailed explanation of the solution.

I have a convex polygon and I want to divide into 4 equal parts using the two perpedicular splits. Like in a picture. I need s1 = s2 = s3 = s4; I need to get coordinates of point where the lines cross and angle of rotation of polygon. I think that firstly I need […]

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$). For irrational $R$, almost all were roots of quadratics, quartics, and a few sextics that can factor over a square root: $\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7},\sqrt{11}$ (and […]

Why did the ancient Greeks give so much importance to the construction of regular polygons with $n$-sides using only ruler and compass and tried to study for what $n$ was such a construction possible? Until Gauss-Wantzel, this was a famous open problem in Euclidean geometry. Can anyone throw any light on its importance?

My question: Let $n$ points $A_1, A_2,\ldots,A_n$ lie on given circle then show that $\operatorname{Area}(A_1A_2\cdots A_n)$ maximum when $A_1A_2\cdots A_n$ is an $n$-regular polygon.

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