Questions tagged [polygons]

For questions on polygons, a flat shape consisting of straight lines that are joined to form a closed chain or circuit

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A, B, C,and D are consecutive vertices of a regular polygon, so $\frac{1}{AB}$ = $\frac{1}{AC}$ + $\frac{1}{AD}$ how many sides does the polygon have? [duplicate]

I let AB = 1, and letting AC = x, and AD = y set up 1 = $\frac{1}{x}$ + $\frac{1}{y}$ , multiplied to get xy = y + x, added 1, and got (x-1)(y-1) = 1 I know AC and AD are equal, but don't know how to ...
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Find the area of ​a regular pentagon as a function of its diagonal

For reference: Calculate the area of ​​a regular pentagon as a function of its diagonal of length $a$. (Answer:$\frac{a^2}{4}\sqrt\frac{25-5\sqrt5}{2}$) My progress: $R$ = radius inscribed circle $...
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36 views

Area of ​convex regular polygons

Problem statement: $a)$ Show that the supreme of the areas of the convex regular polygons with prefixed perimeter p, is given by the area of the corresponding circle with perimeter p. $b)$ Show that ...
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What is the generic name for an $N$-sided polygon whose sides are $a$, $b$, $a$, $b$ and so on? [closed]

Obviously, the number of sides has to be even. If $N = 4$, you have a parallelogram. I guess Regular truncated polygon might be close for $N > 4$, but that only refers to polygons where all the ...
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36 views

Minimum distance between convex polygon vertices on a plane?

I'm facing the following problem: I have a plane with a set of simple convex polygons on top of it. I'm required to prove that however I pick two vertices, one as a starting point and one as a target, ...
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28 views

How much of the surface of these equilateral triangles would be lit?

Consider an equilateral triangle $\Delta ABC$ in 3D with $A=(1,0,0)$, $B=(0,1,0)$ and $C=(0,0,1)$ as well as its mirror image $\Delta A'B'C'$ with $A'=-A$, $B'=-B$ and $C'=-C$. We assume that these ...
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1answer
81 views

Prove that in a $4n$-gon, every other diagonal passes through a common point

Suppose two regular $2n$-gons in the plane, which interesect one another to form a $4n$-gon. Prove that every other diagonal of this $4n$-gon, i.e. $P_{1}P_{2n+1},P_{3}P_{2n+3},...,P_{2n-1}P_{4n-1}$ ...
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Why can triangles only have one shape with three particular side lengths, whereas polygons with more sides can have many possible shapes?

For example, a triangle with side lengths 2, 3, and 1.5 can only have one shape, whereas a parallelogram with side lengths 2,4,2,4 can be a rectangle, or a rhomboid, or you could keep pushing it down ...
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40 views

Approximating $\pi$ with the help of a regular $k$-sided polygon

I'm reading "An Introduction to Computational Physics" by Tao Pang. In it, he writes the following. In general, if the side length of a regular inscribed $k$-sided polygon is denoted as $l_k$...
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74 views

Find the perimeter of a polygon $ABCDEF$

A circle, with a radius of $12$ cm and with the center coinciding with the center of an equilateral triangle with a side of $36$ cm, intersects the sides of the triangle at points $A, B, C, D, E$ and $...
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1answer
67 views

$S$ is bounded if and only if $D=\{\mathbf 0\}$

The set of all recession directions of the polyhedron $S=\{\mathbf x:\mathbf A\mathbf x \le \mathbf b ,\mathbf x \ge \mathbf 0\}$ is equal to the set $D=\{\mathbf d:\mathbf A\mathbf d \le \mathbf 0 ,\...
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1answer
23 views

Rotating all sides of a polygon around its centre such that one of its sides are parallell to the x-axis [closed]

I have a list of vertices in a polygon ((x1,y1),(x2,y2),...,(xn,yn)). I have found the polygons centroid and am able to rotate it using a rotation matrix as explained in this post: Rotate polygon ...
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155 views

Can the perimeter of a convex polygon X be an odd number?

Polygon X is a convex polygon which: is not a triangle has no pair of parallel sides has all vertices with both integer coordinates has sides with a length expressed by a positive integer Can the ...
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1answer
34 views

Recursivley count triangulations of a convex polygon

I am trying to find a recursive number of different triangulations of a convex polygon with $n$ vertices. After some searching I found that the number can be expressed using catalan numbers, this ...
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34 views

A corollary of Ptolemy's theorem [duplicate]

Theorem 1: Consider an equilateral triangle $A_1A_2A_3$ inscribed in a circle. Let $P$ be any point on the circumference. Then $$PA_1+PA_3=PA_2$$ This is easily followed by Ptolemy's theorem. $$PA_1\...
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Combinatorial geometry: vectors in a regular hexagon with the square of the modulus of their sum that is a natural number, which is divisible by 4

We have a regular hexagon with sides of length $n \in \mathbb{N}^*$. On each side, we draw $(n - 1)$ points to split the side into $n$ segments of equal length - $\ell = 1$. We draw all the lines ...
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1answer
55 views

Incircle of polygon tangent to a point

How can we find the largest incircle (not sure if it is still called incircle) tangent to a given point on a side of a polygon? Instead of being tangent to all sides of the polygon, it will be tangent ...
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how to determine that there are 21 vertex types of tilings by regular polygons?

How to determine the 21 vertex types of tilings by regular polygons? ive been searching all through internet to find its exact process to determine the sets of polygons and how they comeup on that ...
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50 views

Convex sided polygon with exterior angles in AP [duplicate]

This question has been asked before, but I have doubts regarding the answer given and being accepted over there : (Link :- Convex n-sided polygons whose exterior angles expressed in degrees are in ...
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46 views

Polygon inside other

Suppose I have two polygons $S_1$ and $S_2.$ Like this: Now I need to check $S_1$ is behind the another polygon or not. I have found one approach in internet. i) Set the plane equation of $S_2(Ax + By ...
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Generalization of Heron's formula for $n$-gons

Recently I knew about Heron's formula for the area of some triangle, and its generalizations to quadrilaterals by Bretschneider's formula. According to Wikipedia there are also generalizations for ...
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How do I trim a polygon to be inside of a square

I want the below question answered so I can code it and help my understanding of how this works so that I can eventually use it as a part of a 3D engine. I am asking it here, because I want to ...
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48 views

Prove that squares split the plane into regions that have boundary equal to some Jordan curve.

In Tao's proof of the Jordan curve theorem in the appendix of his 246A Notes 3 he covers a simple closed curve $\gamma$ with squares of small sidelength and claims that "the boundaries of these ...
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1answer
376 views

Convex n-sided polygons whose exterior angles expressed in degrees are in arithmetic progression

If the exterior angles of a convex n-sided polygon, are all integers, expressed in degrees, are in arithmetic progression, how many values are possible for $n$? The sum of all exterior angles has to ...
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9 views

how would the number of vertex change adding one external point to a convex polygon?

Given a convex polygon $P$ with $n$ vertices, adding a point $q$ in the plane and external to $P$. I can get a new convex polygon $\hat{P} = CH(P\cup \{q\} )$ by finding the convex hull of them. Now ...
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1answer
60 views

How to visualise $1$-gon and $2$-gon for the dihedral groups $D_1$ and $D_2$?

For $n≥3$, the Dihedral groups may be defined as a collection of rotational isometries $r, r^2, \ldots, r^n=e$ and reflection isometries $s, sr, sr^2, \ldots, sr^{n-1}$ satisfying $(sr)^2 =e$ of $n-$...
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21 views

Finding a point that always cuts a polygon into halfs of similar size

I've been trying to solve this problem: Let $P\subset\mathbb{R}^2$ be a closed polygon with area $A(P)$, not necessarily convex. Prove that there is a point $x\in\mathbb{R}^2$ such that every line ...
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29 views

Calculating the area of a hexagon (6-sided entity) alternative

I am kinda curious in something. When looking at a hexagon (6-sided entity) and wanting to calculate the area, could you in theory treat the hexagon as 4 triangles and one square in the middle like ...
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65 views

How does Brownian motion exit a square? Or the harmonic measure on a square.

Consider a circle $\mathcal{C}$ in $\mathbf R^2$ of unit radius and a Brownian motion $\{X_t\}_{t\in \mathbf R^+} $ starting from its center. Let $\mu$ be the measure on $\mathcal{C}$ defined on arcs $...
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2D packing problem - how to optimise/maximise area of a a set of irregular convex polygons within a polygon?

I'm interested in a particular case of this problem: fitting odd-shaped polygons/shapes within the bounds of a rectangle. Say you have 10 sticker designs and you want to fit them all on a sheet of A4 ...
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1answer
61 views

Find vertices coordinates of regular pentagon given one of the coordinates and centre

I need to find the vertex coordinates of a pentagon given that one of the coordinates is $(1,0)$ and that the centre of the pentagon is located at $(0,0)$? This is to be done only by using geometry (...
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2answers
166 views

Non-trig solution for AMC 10 Question

Equiangular hexagon $ABCDEF$ has side lengths $AB=CD=EF=1$ and $BC=DE=FA=r$. The area of $\triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $r$? AMC 10A ...
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39 views

Dihedral group as a subgroup of the symmetric group.

Let $D_n$ be the dihedral group of a regular $n-$polygon. I'd like to write $D_n$ as a subgroup of $S_n$, the set of all permutations of $n$ objects and therefore show why the order of $D_n$ is $2n$. ...
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1answer
172 views

How practical is it to test whether a particular number is polygonal?

Let m be an arbitrary whole number, and d be a whole number value that we wish to deduct sequentially from m, until we find the first example of a result that is non-trivially polygonal as a plane ...
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3answers
84 views

Find Interior Angles of Irregular Symmetrical Polygon

Apologies if this is has an obvious answer, but I've been stuck on this for a bit now. I've been trying to figure out how to make a symmetrical polygon with a base of m length, with n additional sides ...
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Given n nodes sampled uniformly in unit square, does the enclosed area formed by TSP solution has a limit at 0.5 when n approaches infinity

For a TSP of size $n$ Let $(x_i,y_i)$ denote a 2D random variable with $f_i(x,y)$ being its pdf, $i=1,2,\ldots,n$; Let $V_n:\mathbb{R}^{2n}\longmapsto\mathbb{R}^+$ denote a mapping from coordinates ...
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Constructing regular $mn$-gon if $m$-, $n$-gons are constructible

I am studying Stillwell's Elements of Algebra. He mentions the following exercise: If $\gcd(m, n) = 1$ and the regular $m$-gon and the regular $n$-gon are constructible, show that the regular $mn$-...
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Compare polygons shape similarity with noise and different number of vertices

Suppose that I have a reference polygon, $P_1$, and a test polygon, $P_2$. Both are defined by a certain number of ordered points: $P_1$ = {$(x_1, y_1), (x2, y2)...(x_{n1}, y_{n1})$} $P_2$ = {$(x_1, ...
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156 views

Inequality conjecture for convex pentagons

Let $X_1, ..., X_5$ be the vertices of a convex pentagon with perimeter $p$ with its centroid at the origin, satisfying $d(X_i, X_j) < \frac{p}{3}$ for $1 \leq i,j \leq 5$, where $d$ is the ...
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119 views

Distances transformation to polygon boundary for interior vector under translation and rotation

Suppose that we are given A polygon $P$ comprised of $N$ points Two vectors $s_1(x_1,y_1,\theta_1)$ and $s_2(x_2, y_2,\theta_2)$, whose origins are interior to $P$ and Two sets of distances, one for ...
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0answers
38 views

Which topological surfaces' fundamental polygons have opposite sides identified (anti-)parallel?

Any closed topological surface can be constructed from a fundamental polygon with $2n$ sides, with pairs of edges identified according to a specified orientation. Which closed surface results from ...
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39 views

Packing maximum number of identical rectangles in a polygon

Given a 2D polygon, convex or nonconvex, maybe some holes (also small polygons) inside. How can we pack an (approximately) maximum number of identical rectangles in the polygon (the rectangle can not ...
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39 views

A coin strikes the circumference of a circle from within the boundary of the circle at some random angle.

A coin strikes the circumference of a circle from within the boundary of the circle at some random angle theta normal to tangent at that point (where theta is a constant), and keeps moving in the same ...
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1answer
28 views

Can we always find a rectilinear polygon that includes a set of N line segments?

This is a follow-up on my pervious post. Given a set of N rectilinear line segments on 2N distinct points (with integer coordinates) on a square grid. Can we find a rectilinear polygon that passes ...
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1answer
45 views

Can we find a rectilinear polygon that excludes a set of N line segments?

Given a set of N rectilinear line segments on 2N distinct points (with integer coordinates) on a square grid. Can we find a rectilinear polygon that passes through the endpoints of the N line segments ...
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17 views

Smallest circle on hull of a convex polygon

Is there any algorithm that finds the point(s) on the hull of a convex polygon that defines a smallest enclosing circle? So kind of like an algortihm for the smallest enclosing circle problem but only ...
30
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1answer
626 views

Prove that for one vertex of a convex pentagon, the sum of distances to the other four is greater than the perimeter

The problem is from the journal 'Crux Mathematicorum', originally proposed by Paul Erdős and Esther Szekeres for the case of a convex $n$-gon with $n > 5$, and can be found here together with a ...
24
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1answer
229 views

Does a random set of points in the plane contain a large empty convex polygon?

Suppose I choose $n$ points uniformly at random from the unit square $[0,1]\times [0,1]$, obtaining a set of points $S=\{p_1,\ldots, p_n\}\subset [0,1]\times [0,1]$. Then $S$ may contain subsets which ...
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1answer
53 views

Is it guaranteed that the centroid of a convex polygon will be the intersection points of lines connecting opposite vertices?

From Wikipedia, the centroid $\mathbf{C}$ of a finite set of points $\mathbf{x}_1, \mathbf{x}_2, \cdots, \mathbf{x}_k$ in $\mathbb{R}^n$ is: $$ \mathbf{C} = \frac{1}{k} \left( \mathbf{x}_1 + \cdots + \...
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1answer
44 views

Convert area found with Shoelace from pixels to cm

I am working on an app that finds areas of various polygons using the Shoelace theorem. The x and y are in pixels but then the user can set one length in the drawing. For example - a user can draw a ...

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