# Questions tagged [polygons]

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109
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### 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

757
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### 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

138
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
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0
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20
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### 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 ...

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41
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### 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

90
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

53
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

104
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### 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

354
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

63
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

716
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

127
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

129
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

249
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

44
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

61
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

1
answer

135
views

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

In the (arxiv) paper, Exhaustive search of convex pentagons which tile the plane, on page 4 under the proof of Lemma 2, it is said that:
"... We keep a connected component $H_d'$ of $H_{d}$ such ...

1
vote

1
answer

66
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

88
views

### 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

204
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

396
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

575
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 ...

2
votes

1
answer

73
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

143
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

1
answer

155
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 ...

0
votes

0
answers

62
views

### Reference request: realizing a set of numbers as interior angles of a convex n-gons

Let $\{a_1,...,a_n\}$ be a set of positive numbers. Recently, a smart kid asked me how to decide if there is a convex polygon $A_1A_2...A_n$ such that the interior angle at $A_i$ is $a_i$ degrees. Let'...

4
votes

1
answer

182
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

183
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 ...

1
vote

1
answer

227
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 ...

1
vote

1
answer

240
views

### A formula for the area of bicentric quadrilateral

Can you provide a proof for the claim given below? The following claim is inspired by Harcourt's theorem and can be seen as its generalization to quadrilaterals.
Claim. Given bicentric quadrilateral $...

6
votes

0
answers

94
views

### Zero-area-free embedding of points on the grid

Given $n$, I am looking for the smallest $m$ such that there is an $n$ element subset $S$ of the $m\times m$ grid (i.e., $n$ points with integer coordinates in $[0,m]^2$) such that no matter how one ...

3
votes

0
answers

122
views

### Illuminating a just-barely irrational polygon

As has been discussed earlier on MO,1,2
recently an impressive advance was proved concerning
internally illuminating a mirrored polygon.
Here is the result:
Corollary 3. Let $P$ be a rational polygon....

0
votes

0
answers

24
views

### Maintain the area of a polygon when offsetting one side

I have an irregular polygon with the a specific area (area_red). How can I get the (parallel) offset value (y) of n selected sides in order to maintain the same area (area _red = area_green) when (...

1
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0
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54
views

### Is the mean center of the vertices of a convex polygon always inside the polygon? [closed]

As simple as that. I'm doing an R program where I need to order clockwise a bunch of points that describe a regular polygon and to do that I figured I could find a point inside, change to polar ...

8
votes

1
answer

168
views

### Counting polygons in arrangements

For an arrangement of lines $\cal{A}$ in the plane, an
inducing polygon $P$ is a simple polygon satisfying:
(a) every edge $e$ of $P$ lies on some line $\ell$ of $\cal{A}$, and
(b) every line $\ell \...

3
votes

1
answer

702
views

### Brother of Japanese theorem for cyclic quadrilaterals

I am looking for a proof of a like result as follows and Higher-dimensional generalizations?
Let $A, B, C, D$ be four point with lengths of $AB, BC, CD, DA$ are $a, b, c, d$ respectively. Let $F \in ...

4
votes

1
answer

321
views

### Construct closed chain of $k$-gon around $n$ points-$n, k$ are odd primes number

Question 1: I am looking for a proof of the conjectures 1, 2, 3 as follows?
Question 2: In conjecture 3, in general case, I can not give a formula of $X$. But I think, If $n, k$ are odd primes number ...

4
votes

1
answer

205
views

### A closed chain of $2n+1$-gon around $2n+1$-points

I posed a generalization of Theorem 3.2 In my paper
Conjecture: Let $P_1, P_2,....,P_{2n+1}$ and $O$ be $2n+2$ points in plane. Construct a chain $2n+1$ regular ${2n+1}$-gons $A_{1\;1}A_{1\;2}...A_{1\;...

1
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0
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73
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### How can construct the equilateral $A''B''C''$ such that area of $A''B''C''$ is biggest

Let $ABC$ be arbitrary triangle in a plane. Let $A'B'C'$ and $A''B''C''$ be two equilateral triangles such that $A \in B'C'$, $B \in C'A'$, $C \in A'B'$ and $A \in B''C''$, $B \in C''A''$, $C \in A''B'...

3
votes

1
answer

207
views

### How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle

How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle?
See also:
Malfatti circles

3
votes

0
answers

159
views

### Hypothesis: An injection from polygons into $SO(2) \times S_n$

I have stumbled upon a possible representation of polygons by a concise description of their behaviour under rotation. I would like to know more about it and, obviously, if it is actually a bijection. ...

11
votes

2
answers

619
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### Strange formula for area of a convex polygon

Consider a convex $n-$gon in $\mathbb{R}^2$ with sides contained in the lines $y=k_ix+b_i, 1\leq i\leq n.$ Then its area equals to
$$
S=\frac{1}{2}\sum_{i=1}^{n} \frac{(b_{i+1}-b_i)^2}{k_{i+1}-k_i}.
$$...

0
votes

1
answer

123
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### Point generation in polygon

I know about the Halton sequence. But so far I can’t find the formulas by which points are generated. Also worried is the question Halton sequence generates points only in the rectangle? Or can I ...

1
vote

1
answer

192
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### Can we split a polygon in half (vertex-wise) by a diagonal, but with a constant maximum difference?

This is a followup to my last question - Can we prove that simple polygons can always be split in half (vertex-wise) by diagonals?
Is there a constant natural number K for which the following is true:...

0
votes

1
answer

67
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### Can we prove that simple polygons can always be split in half (vertex-wise) by diagonals?

Can we prove that for any simple polygon with more than 3 vertices there always exists a diagonal which:
is inside the polygon
doesn't intersect with any edges
splits the polygon in two polygons in ...

0
votes

0
answers

142
views

### Automorphism group of the cycle graph with $k$ diagonals

Let $C_n$ be the cycle graph on $n$ vertices. Then $Aut(C_n) \cong D_{2n}$ the dihedral group of order $n$.
Now, let $C_{n,i}$ be the cycle graph with $i$ many diagonals connecting opposite vertices ...

2
votes

0
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### Complexity of existence of simple polygonalization with prescribed area?

This is a followup on my previous question. Fekete proved the NP-completeness of deciding the existence of simple polygonalization with minimum (or maximum) enclosed area (simple polygonalization is ...

1
vote

1
answer

99
views

### Projections of particular simplex yielding boundary of a regular polygon?

What is the maximum $m$ such that there is a simplex with $n$ vertex points in $n-1$ dimensions whose projection yields boundary of a regular $m$-gon on $2D$ plane?

2
votes

1
answer

327
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### Build reversed No-Fit-Polygon

I need some robust algorithm to optimally fit one non-convex polygon into another. The destination one can contain holes.
Recently I found scholarly articles on this subject:
One of them describes ...

5
votes

1
answer

204
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### Generalizations of the "Curious Tiger" Polygon

I actually don't know, whether the polygon I describe here already has name, but let me explain the problem, that is solved by the polygon, with a little story:
Imagine a flat terrain with bushes of ...