# Questions tagged [polygons]

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92
questions

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### 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 ...

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71 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 ...

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175 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,
$$
\...

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

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520 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 ...

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50 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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130 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\...

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136 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 ...

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60 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'...

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124 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 ...

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134 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 ...

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161 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 ...

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206 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 $...

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92 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 ...

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77 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....

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20 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 (...

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47 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 ...

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129 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 \...

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634 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 ...

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311 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 ...

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194 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\;...

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66 views

### 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'...

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165 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

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155 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. ...

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576 views

### 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
$$
S=\frac{1}{2}\sum_{i=1}^{n} \frac{(b_{i+1}-b_i)^2}{k_{i+1}-k_i}.
$$...

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112 views

### 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 ...

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104 views

### 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:...

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58 views

### 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 ...

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103 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 ...

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50 views

### 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 ...

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95 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?

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228 views

### 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 ...

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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 ...

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105 views

### Chain rotation of a point

Let $n$ be a positive integer number and $P$ be a point in a plane. Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ points in the plane, we take modulo $m$ for $A_j$ (it is mean $A_{m+i}=A_{i}$ for $i=1, 2, \...

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154 views

### Complexity of 2D-Minkowski sum of non-convex polygons

I have read that the complexity of computing the Minkowski-Sum of $2$ non-convex polygons (through convex decomposition) is $O(m^2 n^2)$, where $m$ and $n$ is the number of vertices of each polygon. ...

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155 views

### How many points appear in the plane when the chain of n-gons is close?

Let $A_{11}A_{12}\cdots A_{1n}$ be a regular $n$ polygon, we call $A_{11}A_{12}\cdots A_{1n}$ is the $1st-n-gons$. Now we construct the $2nd-n-gon$ based two condition as follows:
$2nd-n-gons$ is ...

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288 views

### In arbitrary cyclic polygon then $\sum_{i<j} x_{ij}^\alpha \ge \sum_{i<j} y_{ij}^\alpha $

I am looking for a proof of the inequality as follows:
Conjecture: Let $A_1A_2....A_n$ be the regular polygon incribed a circle $(O)$. Let $B_1B_2....B_n$ be a polygon incribed the circle $(O)...

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### An inequality related to area and sidelengths of a polygon $Area(A_1A_2…A_n) \le \frac{1}{n}cotg{\frac{\pi}{n}} \sum_{i=1}^nA_iA_{i+1}^2$

I am looking for a proof (or a reference) of an inequality related to a rea and the sidelengths of a polygon as follows:
Let $A_1A_2...A_n$ be arbitrary polygon, then:
$$Area(A_1A_2....A_n) \...

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406 views

### Fitting one Polygon in another

I have two Polygons A and B and I want to find the position, rotation and scale of B, so it fits into A and has the maximum Area possible. Also both can be concave.
I did some research but couldn't ...

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### What is the name of the 65537-gon? [closed]

I know the name of the heptadecagon (17 sides) and the diacosipentacontaheptagon (257 sides). But what is the name of the polygon with 65537 sides? I am unable to figure it.

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### Reordering vertices of a polygon

Let $Q,Q'$ be two planar polygons with the same number $n>3$ of vertices. There is a correspondence between vertices of $Q$ and $Q'$: to any vertex $z$ of $Q$ corresponds a unique vertex $z'$ of $Q'...

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### Can one “hear” the shape of a polygon via external reflections?

This question is a rough analog of Kac's "Can One Hear the Shape of a Drum?"
A closer analog is the recent "Bounce Theorem" that says, roughly, the shape of a polygon is determined by its billiard-...

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112 views

### Is it possible to dissect a regular polygon into mirrored-symmetric pieces?

Q1. Planar regular triangle is dissected into three congruent pieces, each of them having no symmetry axis.
Can it be so, that one of these pieces is a mirrored (and then rotated) copy of some other ...

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### Rectangular Newton polygon of a Jacobian pair

Let $p,q \in k[x,y]$, $k$ is a field of characteristic zero.
By definition, $p,q$ is a Jacobian pair if their Jacobian is invertible in $k[x,y]$, namely, $p_xq_y-p_yq_x \in k^*$, and $p,q$ is an ...

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858 views

### Maximum area of the intersection of a parallelogram and a triangle

How large can the intersection of a parallelogram (or a square, if you prefer) with a triangle be, if each of them is of unit area? It is easy to see that the intersection can be of area 3/4 – is this ...

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275 views

### Problem on distances in a polygon

In $\mathbb{R}^2$ consider a square (call it $S$) and three triangles (one acute $T_2$ and two obtuse $T_1$ and $T_3$) such that each triangle shares one different side with the square and the ...

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**1**answer

1k views

### decompose rectilinear polygons into rectangles with minimum internal borders

I have a number of rectangles as shown in the attached diagram (labelled original) and I want to merge them together in a way (labelled Result ) to get the minimum internal borders. The reason I want ...

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680 views

### Algo for covering maximum surface of a polygon with rectangles

I'm looking to an algorithm to covering maximum surface of a polygon with rectangles. Rectangles have to have a specific width, a rectangle can't overlap an other one and each one has to fit in the ...

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**1**answer

678 views

### Three homothetic centers are collinear

I am looking a proof for the problem as follows:
Let a convex hexagon, such that its principal diagonals are concurrent. For each side of the hexagon, extend the adjacent sides to their ...

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**0**answers

326 views

### Determining convexity of a polygon from its Fourier coefficients

Consider an $n$-sided polygonal curve in the plane, represented by an ordered set of points $(x_0, x_1, \ldots, x_{n-1})$; line segments connect consecutive points and also $x_{n-1}$ to $x_0$. It is ...