# Questions tagged [polygons]

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### Maximum coverage of an orthogonal polygon using $k$ rectangles

I have an orthogonal polygon (all edges are horizontal or vertical) which is convex (no holes in any row of column of the polygon). I would like to cover as much as possible of this orthogonal polygon ...
118 views

### Incenter-of-mass of a polygon

"Circumcenter of mass" is a natural generalization of circumcenter to non-cyclic polygons. CCM(P) can be defined as the weighted average of the circumcenters of the triangles in any ...
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### Triangle centers formed a rectangle associated with a convex cyclic quadrilateral

Similarly Japanese theorem for cyclic quadrilaterals, Napoleon theorem, Thébault's theorem, I found a result as follows and I am looking for a proof that: Let $ABCD$ be a convex cyclic quadrilateral. ...
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### Do cut-length-minimizing equidissections exist?

Suppose $A,B$ are polygons of equal area. By the Wallace-Bolyai-Gerwien theorem, $A$ and $B$ are equidissectable: we can make finitely many straight-line cuts in $A$ and rearrange the resulting pieces ...
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### Regular $n$-gon with diagonals: bounds on area of largest cell?

Consider a regular $n$-gon of side length $1$ with diagonals. Here is an example with $n=11$ (from geogebra applet). I've been trying to find, in terms of $n$, bounds on the area of the largest cell, ...
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### How to characterize the regularity of a polygon?

In my research, I've recently started to play with Voronoi tessellations. I currently have a Python code that creates the tessellation and I am trying to color the polygonal regions according to their ...
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### Happy ending problem - why not a proof by induction? (cont)

After sharing ideas on this post, I have been thinking for some time on the problem, and I think that a possible way to prove the Erdös-Szekeres conjecture could be structured as follows: Consider ...
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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 ...
122 views

### 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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1 vote
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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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