# Questions tagged [polygons]

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### A regular $n$-gon contains a regular $m$-gon, with $n,m$ coprime, no sides coinciding. What is the maximum number of contact points between them?

A regular $n$-gon contains a regular $m$-gon, where $n$ and $m$ are coprime, with no sides coinciding. What is the maximum number of contact points between the $n$-gon and the $m$-gon? (I'm not ...
1 vote
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### All the regular $n$-gons are nested tightly around a unit circle. How to order them to minimize the outer radius, and what is that minimum radius?

Let $u_1,u_2,u_3,\dots$ be a permutation of the integers greater than $2$. A unit circle is in a regular $u_1$-gon, which is a regular $u_2$-gon, which is in a regular $u_3$-gon, ad infinitum. Each ...
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### Minimum reflection paths in a mirror polygon

Let $P$ be a simple, orthogonal polygon of $n$ edges, i.e., one whose edges meet at right angles, and is non-self-intersecting; also known as a rectilinear polygon. Treat every edge of $P$ as a ...
218 views

### Is there a bicyclic irregular pentagon in integers?

Is there a bicyclic irregular pentagon in integers, i.e. is there a pentagon, the length of each side is integer and unique such that it has a circumcircle and an inner circle as well? If it does ...
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### Necessary and sufficient condition for quadrilateral to be cyclic

Can you provide a proof for the following proposition: Proposition. Given any quadrilateral $ABCD$. Let $P,Q,R,S$ be nine-point centers of triangles $\triangle ABD$,$\triangle ABC$,$\triangle BCD$ ...
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### Acute triangles in "obtuse" polygons?

Let $P$ be a convex polygon. Suppose every interior angle of $P$ is obtuse. Is it always the case that there exist three vertices $p, q, r$ of $P$ such that $\triangle pqr$ is acute? I conjecture ...
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### Collinearity of three significant points of bicentric pentagon

Can you provide a proof for the following claim? Claim. Given bicentric pentagon. Consider the triangle whose sides are two diagonals drawn from the same vertex and side of pentagon opposite from ...
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### Does there exist an isometry between a regular polygon and a circle?

In order to define the question in a meaningful fashion, I am referring to a smooth manifold $\mathcal{M}$ within an $\epsilon$-neighborhood of a regular polygon $\mathcal{P}$ satisfying \max\{\|x-p\...
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