# Questions tagged [polygons]

The polygons tag has no usage guidance.

131
questions

2
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0
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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 ...

3
votes

1
answer

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 ...

3
votes

1
answer

135
views

### 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.
...

6
votes

1
answer

207
views

### Inscribing one regular polygon in another

Say that one polygon $P$ is inscribed in another one $Q$, if $P$ is contained entirely in (the interior and boundary of) $Q$ and every vertex of $P$ lies on an edge of $Q$. It's clear that a regular $...

0
votes

1
answer

168
views

### Density of the set of convex polygons in the Banach-Mazur distance

Is the set of convex polygons dense in the set of convex domains in $\mathbb{R}^2$, for the Banach-Mazur distance?
Any insight for a negative or positive answer is very much welcome!

5
votes

0
answers

180
views

### Question about $n$ random points in a regular polygon, and a limiting probability

Suppose we choose $n$ uniformly random points in a disk, then draw the smallest circle that encloses all of those points. There is evidence suggesting that the probability that the enclosing circle is ...

4
votes

1
answer

321
views

### Billiard circuits in pentagons

A billiard circuit in a convex $n$-gon is a closed billiard path
of $n$ segments reflecting from consecutive edges of the polygon.
Every regular $n$-gon has such a billiard circuit:
Recently a ...

1
vote

0
answers

102
views

### 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

0
answers

92
views

### 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 ...

10
votes

0
answers

166
views

### 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 ...

7
votes

1
answer

491
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 ...

0
votes

1
answer

49
views

### What is the most dense sample for which the Crust algorithm returns an incorrect polygonal reconstruction?

The Crust algorithm by Amenta, Bern, and Eppstein computes a polygonal reconstruction of a smooth curve $C$ without boundary from a discrete set of sample points $S$. It is known that if $S$ is an a $\...

10
votes

0
answers

197
views

### 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 ...

14
votes

0
answers

266
views

### 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, ...

14
votes

2
answers

732
views

### 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 ...

4
votes

0
answers

190
views

### 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 ...

3
votes

1
answer

408
views

### Generalization of some plane geometry theorems

Conjecture: Let $A_1, A_2,\dotsc,A_n$; $B_1, B_2,\dotsc,B_n$ and $C_1, C_2,\dotsc,C_n$ be $3n$ points in the plane such that $\angle{\overrightarrow{A_iB_i}, \overrightarrow{A_{i+1}B_{i+1}}}=\frac{2\...

1
vote

0
answers

148
views

### Stronger conjectured inequality for area of a polygon

Four years ago, I proposed an inequality related to area and sides of a polygon.
After computer checking, I conjecture that the previous inequality can be strengthened as follows:
Let $A_1A_2\cdots ...

2
votes

1
answer

389
views

### Double integral in a polygon domain

I want to compute a integral of a polynomial $f(x, y)$ over a polygon domain $D$ of $n$ sides.
$$
I(f) = \int_{D} f(x, \ y) \ dx \ dy
$$
The vertex of this polygon are
$$\vec{p}_{i} = (x_i, \ y_i) \ \ ...

1
vote

1
answer

132
views

### Rotational invariance assumed, what is the number of $r$-sided simple polygons that can be inscribed into an $n$-sided regular polygon?

When I say that an $r$-sided simple (i.e., not self-intersecting) polygon is inscribed into an $n$-sided regular polygon, I mean that every vertex of the simple $r$-gon is also a vertex of the regular ...

5
votes

2
answers

563
views

### First Dirichlet eigenvalue on regular polygons

Assume $\lambda(P)$ is the first Dirichlet eigenvalue of a regular polygon $P$. Let $u$ be the corresponding eigenfunction, normalized by $\|u\|_{L^2(P)}=1$, and $\partial_{\nu}u$ be its normal ...

0
votes

0
answers

108
views

### Explicit equation for border of the Minkowski sum of sets

Assume we have sets of the form
$$
M_j = \{x\in\mathbb{R}^d : f_j(x) \le 0,x \ge 0\}
$$
where $x\ge 0$ means $x_i \ge 0 \quad \forall i=1,\dots, d$.
Goal
I am looking for an (explicit) representation ...

15
votes

4
answers

810
views

### Unlinked interlocking planar polygons

Let $P$ and $Q$ be the boundary segments of two planar simple polygons.
View these boundaries as rigid wires.
Fix $Q$ in, say, the $xy$-plane, and imagine $P$ arranged in $\mathbb{R}^3$ so that $P$ ...

5
votes

1
answer

342
views

### Inserting points into a polygon

I want to insert $n$ points into arbitrary polygon $P$ described by ordered list of its vertexes $v_1, v_2, ..., v_m$. Each inserted point must distanced from the others on distance at least $d$. In ...

1
vote

0
answers

30
views

### Separability of graph component embeddings

I have an undirected graph. I also have an embedding of the graph in $\mathbb{R}^3$. Assume that the graph has 2 connected components. I want to know whether there exists a plane (or better - any ...

1
vote

0
answers

60
views

### To extend the Steiner-Lehmus theorem

The Steiner Lehmus theorem (https://en.wikipedia.org/wiki/Steiner%E2%80%93Lehmus_theorem) states: Every triangle with two angle bisectors of equal lengths is isosceles.
Question: What could one say ...

0
votes

1
answer

393
views

### The dimension of the normal cone of a face in a polytope

Let $P$ is a polytope in $\mathbb{R}^n$ if $F$ is one of its faces of dimension $d$ then the dimension of its normal cone $\mathcal{N}(F)$ is $n-d$.\
This seems to be intuitively obvious but I can't ...

3
votes

1
answer

64
views

### Scissor congruence for foliated polygons

Given two polygons of equal area with horizontal foliations, can one describe the obstruction (if there is any but I suspect the answer to be yes) to scissor-equivalence respecting the horizontal ...

1
vote

1
answer

124
views

### Is there a non-trivial, exact analytic (symbolic) conformal map from some polygon to some rectangle?

I'm looking for any example of a conformal map $m: P \to Q$ where $P$ is some polygon of at least $4$ sides, $Q$ is a rectangle, and and $m$ is not linear (so $P$ and $Q$ are not merely scaled, ...

6
votes

1
answer

603
views

### Sum of the lengths of all diagonals in a regular polygon

I believe that the sum of the lengths of diagonals of a regular polygon ($n$-gon) is always greater than or equal to any other irregular polygon ($n$-gon) inscribed in a circle..
For example for a 4-...

1
vote

1
answer

103
views

### Collinearity in tangential pentagon [closed]

I am looking for a proof of the following claim:
Given tangential pentagon. Touching point of the incircle and the side of the pentagon,the vertex opposite to that side and the intersection point of ...

2
votes

0
answers

905
views

### Happy ending problem – Why not a proof by induction?

I have been thinking for a while on the happy ending problem, looking for approaches to attack the Erdős–Szekeres conjecture: the smallest number of points for which any general position arrangement ...

1
vote

1
answer

314
views

### High-dimensional polytopes

I have two questions regarding polytopes in high dimensions, $\mathbb{R}^d$ where $d > 3$, that I could not find resources for on the web for. Suppose I have a polytope that is non-convex:
How can ...

3
votes

1
answer

157
views

### The product of the lengths of two line segments that belong to Newton line [closed]

I am looking for the proof of the following claim:
Consider a family of bicentric quadrilaterals with the same inradius length and the same distance between incenter and circumcenter. Denote by $P$ ...

3
votes

1
answer

285
views

### Need help with finding all angles of 11 sided 3D object [closed]

Question: I'm an artist trying to build a hendecahedron for a project (Image below to see the shape). This object consists of 5 pentagons at the base, 1 pentagon on the bottom, then 5 quadrilaterals ...

1
vote

1
answer

50
views

### On triangulations and "coverage" of circumcircles

Let $P$ be a convex quadrilateral defined by four vertices $a$, $b$, $c$, and $d$. Suppose that the circumcircle of $\triangle abd$ contains $c$.* Let $D(\triangle abc)$ to denote the area enclosed by ...

2
votes

0
answers

80
views

### Principal diagonals of octagon meet in a single point

Can you provide a proof for the following claim:
Claim. Given octagon circumscribed about an ellipse. If the vertices of the octagon lie on another ellipse then its principal diagonals meet in a ...

3
votes

2
answers

271
views

### Construct by compactness (Pentagonal tiling – Rao paper)

In the (arXiv) paper, Exhaustive search of convex pentagons which tile the plane by Michael Rao, on page 4 under the proof of Lemma 2, it is said that:
"… We keep a connected component $H_d'$ of $...

1
vote

1
answer

84
views

### Equal products of triangle areas

Can you prove the following claim:
Claim. Given hexagon circumscribed about an ellipse. Let $A_1,A_2,A_3,A_4,A_5,A_6$ be the vertices of the hexagon and let $B$ be the intersection point of its ...

3
votes

1
answer

103
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### Equal sums of line segments

I would like to see a proof of the following
Claim. Let $A_1,A_2,A_3,A_4,A_5$ be vertices of bicentric pentagon. Let $B_1$ be the intersection point of $A_1A_3$ and $A_2A_5$, $B_2$ the intersection ...

1
vote

1
answer

227
views

### Partitioning a convex $n$-polygon

Let $P$ be a convex polygon with $n\ge3$ vertices. Let $\mathcal{Z}_K(P)$ be a partition of the polygon into $K$ polygonal (but not necessarily convex) parts whose interiors are pairwise disjoint,
$$
\...

6
votes

4
answers

562
views

### 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 $\triangle ABD$,$\triangle ABC$,$\triangle BCD$ ...

15
votes

1
answer

612
views

### 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 ...

3
votes

1
answer

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 ...

1
vote

0
answers

178
views

### 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\...

3
votes

2
answers

205
views

### Generalizing Pythagorean Theorem: equations defining edges of a (convex) $n$-gon with $n-2$ prescribed angles?

Let $n\geq 3$ be an integer and $0<\alpha_1, \dots ,\alpha_{n-2}<1$. Let's say a tuple of positive numbers $(e_1,\dots, e_n)$ is nice if there is a convex $n$-gon $A_1\dots A_n$ such that $\hat ...

4
votes

1
answer

309
views

### Collinearity in bicentric polygons

Can you provide a proofs for the following two claims?
Claim 1. The circumcenter, the incenter, and the intersection of the principal diagonals in a bicentric even-sided polygon are collinear.
Claim ...

6
votes

1
answer

222
views

### Necessary and sufficient condition for tangential polygon to be cyclic

Can you prove or disprove the following claim?
Claim. Let $A_1,A_2, \ldots ,A_n$ be the vertices of an $n$-sided tangential polygon and let $B_1,B_2, \ldots ,B_n$ be the contact points of the ...

8
votes

0
answers

214
views

### Compass and straightedge construction of Poncelet polygons

Gauss–Wantzel theorem states that
A regular n-gon is constructible with straightedge and compass if and only if $n = 2^kp_1p_2...p_t$, where $p_i$'s are distinct Fermat primes (A Fermat prime is a ...

1
vote

1
answer

311
views

### A generalization of Harcourt's theorem

This question is closely related to my previous question.
Can you prove the claim given below? The following claim is a conjectured generalization of Harcourt's theorem.
Claim. Let $A_1,A_2 \ldots ...