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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Finding area of hypocycloids (without integration)

I have been trying to find the area of hypocycloids, I understand how to do it with integration. But the thing is I wanna find some other method for finding its area. In one of the sites online, I ...
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1answer
1k views

IMO 2016 Problem 3

Let $P = A_1 A_2 \cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $...
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31 views

finding farthest vertices to a point in n-dimensional hypercube

I am an engineer not a mathematician. So, please accept my apology if this is a very simple problem for this site. Given a coordinate $A(x_{1A},x_{2A},...,x_{nA})$ and a value d in n-dimension space. ...
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1answer
44 views

Inscribing convex polygon within simple polyon

Suppose you are given a simple (but not necessarily convex) polygon $C$ and a point $p$ inside this polygon. I have a particular way of inscribing a convex polygon $I$ within $P$, and I would like to ...
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1answer
77 views

How many pairs of diagonals of of a odd sided regular polygon intersect within the interior the polygon?

How many pairs of diagonals of a $2n+1$ sided regular polygon intersect within the interior of the polygon? (By interior I mean it shouldn't intersect on the vertex) For a triangle there is no ...
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1answer
55 views

What is the probability that the center of a odd sided regular polygon lies inside a triangle formed by the vertices of the polygon?

Three distinct vertices are choose at random from the vertices of a given regular polygon of $2n+1$ sides. Each of those three vertices will determine a triangle. What is the probability that the ...
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1answer
10 views

Find point where a line of multiple vertices overlaps itself

Since I'm not familiar with a lot of mathematical terminology, I will explain this problem with a little story. Imagine you and your friend Anne have a piece of string each, and place it on a ...
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1answer
104 views

Maximizing area of a pentagon

Suppose $a,b,c,d,e$ are pairwise distinct positive integers. Consider a pentagon with sides $a,b,c,d,e$ and with angles maximizing its area (we assume that a pentagon with such sides exists). It is ...
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1answer
23 views

Does the Ray Casting Algorithm works in Poincare's Disk to detect if point is inside Polygone?

As the Ray Casting Algorithm looks to me like a geometric construction on geodesics, and geodesics are redefined in Poincare's Disk, I feel this method would also work in hyperbolic geometry. Is this ...
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2answers
37 views

The sum of the perimeter of regular polygons inscribed inside of regular polygons

This is a question combining number theory and geometry. I am asking it purely from curiosity, but I think it might be a useful and interesting question. Start with an equilateral triangle of ...
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38 views

split a rectangle with triangles into polygons as uniformly as possible

Given a rectangle $A$ and $n$ triangles $\{B_1,B_2,...,B_n\}$, I put the triangles inside $A$, at least one vertex of each triangle is not outside $A$ (inside $A$ or on the edge of $A$). So that A is ...
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An odd equivalence relation

Consider polygons inscribed in a circle of unit radius. Call two such polygons "equivalent" if their areas are the same and the sums of squares of the lengths of their edges are the same. Two angles ...
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1answer
47 views

Inscribed Shape on Circle given Specific Edges

How would you find the vertices (corners' position) of a shape that inscribes a circle of adjustable radius, given a set of edges? Angles of polygon are not fixed, but edges are. A few examples: ...
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1answer
28 views

Is a shape 'polarizable'?

Given a point $p$ inside a shape $S$ described as an $n$-vertex polygon, let us say that $S$ is polar with respect to $p$ if S can be described by a polar equation $r(\theta)$ with $p$ as the origin. ...
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3answers
94 views

Dodecagon Area Question

The distance between two opposite vertices of the dodecagon is 2. Find the area of the dodecagon. Is there any way to do this without trigonometry? And could you include a proof also? :O
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2answers
37 views

For a regular polygon with sides of length $l$, prove that all points within $l$ from a vertex lie on an incident edge

I am trying to prove that all the isometries of a regular polygon that map the polygon back onto itself must map vertices to vertices. I nearly have the proof, but I need to prove one more statement: ...
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1answer
73 views

How can polygon faces be calculated from edges?

In a 2 dimensional plane [given that I know the vertices of the start and end of a number of connected lines], how can I calculate the faces [enclosed by those lines](and there respective vertices) ...
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24 views

What is the name of a “polygon” with piece-wise polynomial boundary?

I would like to know if somebody knows the name of these objects. Given a set of $N$ vertices $\{(x_i, y_i)\}_{i=1,\ldots,N}$ (points in $\mathbb{R}^2$) we create a closed curve, defined piece-wise, ...
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19 views

Using GPS coordinates in trillateration

for a project we need to find a certain position. The info we have : 3 surrounding positions and the distance between those positions and the point we are looking for. We've got a setup like this ...
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1answer
64 views

How do you calculate the smallest cycle possible for a given tile shape?

If you connect together a bunch of regular hexagons, they neatly fit together (as everyone knows), each tile having six neighbors. Making a graph of the connectivity of these tiles, you can see that ...
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34 views

What's the difference between the Minkowski difference of $A$ and $B$ and the Minkowski sum of $A$ and $-B$?

In the book Computational Geometry, Algorithms and Applications from de Berg, van Kreveld, Overmars and schwarzkopf, I read the following in chapter 13.3 on Minkowski sums: Sometimes $ P \oplus(-R(...
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1answer
31 views

The sum of distances from the sides of a regular polygon to an interior point is a constant

Let there be a regular polygon of $n$ sides. Assume there is a point $P$ inside the polygon, then prove that $$a_1 + a_2 + a_3 + \cdots + a_n= \text{constant}$$ where $a_i$ is the distance of ...
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2answers
38 views

Can a polygon with an infinite number of sides be viewed as a line?

The inner angles of a polygon approach 180º as the number of sides (N) of the polygon increases. So, if N approaches infinity, we would have a circle. But... At infinity, we would also have a set of ...
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3answers
61 views

A simple proof that a polygon circumscribing a circle overestimates its perimeter

Looking at the picture below, it's easy to see why the perimeter of a polygon inscribed in a circle is an underestimation of the circle's perimeter. This follows from the triangle inequality: Any side ...
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0answers
48 views

Number of quadrilaterals formed out of N points.

How can I calculate the number of $4$-vertex convex polygons (quadrilaterals) that can be formed out of $n$ given points, where $n \geq 4$? Note: Points can be collinear. So triangles with $3$ sides ...
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can a convex polygon have only one boundary point at locally maximum distance from its centroid?

It's easy to see that given any convex polygon P and any point c in its interior, there is at least one point m on the boundary of P at locally maximum distance from c: simply choose m to be a vertex ...
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2answers
2k views

Can a polygon be one dimensional?

When looking up the definition of polygon, Wikipedia tells me: In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing ...
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18 views

Does it make sense to compare sets (polygons) with different dimensions?

In the context of integer programming, I am considering 3 different linear models for a given problem. The goal is to determine which formulation is the tightest, that is, the one that gives the least ...
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2answers
60 views

How would the volume of a frustum with irregular polygon area be calculated?

I want to calculate the volume of this shape, it's basically a frustum with an irregular polygon base. The bottom area $A_1$, the height of the frustum shape $h$,the sideways distance between $A_1$ ...
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3answers
133 views

Closed form for the area of a convex cyclic n-gon, given the set of edge lengths

Let's say we are given a set of positive reals, and we're told that these are the edges of a convex cyclic $n$-gon, and we must compute it's area. For $n = 3$ there is the famous Heron's formula: $$...
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2answers
79 views

Prove that $\frac{ a^2 +b^2 +c^2}{d^2}$ is always greater than $\frac{1}{3}$

Prove that if $a,b,c$ and $d$ are the sides of a quadrilateral ,then the value of $\frac{ a^2 +b^2 +c^2}{d^2}$ is always greater than $\frac{1}{3}$ Could someone please give me hint to solve this ...
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1answer
33 views

Determine whether a polygon is convex based on its vertices.

We have a polygon $A_1A_2\ldots A_k \subset \Bbb{R^2}$ with the coordinates: $$A_1 = (x_1, y_1)$$ $$A_2 = (x_2, y_2)$$ $$\vdots$$ $$A_k = (x_k, y_k)$$ Is there any way to determine whether or not ...
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0answers
102 views

Progressive packings in a convex shape

Take a shape, and scale it by 1 to $n$. For a tiny set of tightly related shapes, such as isosceles right triangles with shortest sides 1 and sqrt(2), scale the set of shapes by 1 to $n$. What is ...
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Maximum number of regular polynom projections of a polyhedron

It is trivial that a cube has both a square and a regular hexagonal projection. We can also easily construct a polyhedron with three perpendicular projection, which are different regular polygons. ...
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1answer
80 views

Regular polygon made from the midpoints of the sides of a polygon

For what natural numbers n is it true that whenever the midpoints of the sides of a convex n-sided polygon $K_n$ form a regular n-gon, then $K_n$ itself is also a regular n-gon? I know it is true for ...
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2answers
44 views

Polygon with curved sides, and higher-dimensional generalizations

I am trying to find references about generalizations of polygons with non-straight sides. I am interested in both the convex and non-convex cases, and particularly in polynomial boundaries, and ...
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2answers
25 views

Are there alternatives to polygons in mathematical (computational) modelling?

So polygons are pretty standard in computer graphics, but from a mathematical perspective, one'd expect something more refined and sophisticated to be possible right? Polygons are not very ...
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51 views

Average Perimeter With n Points on the Unit Circle

A couple days ago, a friend challenged me to solve a problem: You have N vertices, each randomly placed on the edge of a unit circle. What is the formula (given N) that yields the average perimeter ...
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3answers
66 views

Is there a regular hexagon with integral corners?

I'm looking for a regular hexagon in $\mathbb{R}^2$, whose corners are integral, i.e. the coordinates are integers. The hexagon cannot lie "flat" (with upper and lower line segments horizontal), ...
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1answer
66 views

Find if 4 lines form a Quadrilateral in 2D space

How can I know if 4 lines form a Quadrilateral in 2D space? And how would I obtain the corners? (in clockwise order starting with the top left corner) Note that lines are formed by 2 points in my ...
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1answer
47 views

Center of gravity of right angle trapezoid vs rectangle

Could you please help me to find the center of gravity of a trapezoid and a rectangle with the following measures? Where is the center of gravity in these two shapes? Trapezoid: a: $20$cm b: $17$ ...
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1answer
30 views

Formula for vertices of a Polygon with only 1 vertex at the top and y-axis symmetric

I'm trying to find the formula for the vertices of a polygon with n-sides such that there is always only 1 vertex at the top and the polygon is symmetric with respect to the y-axis... so generally ...
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2answers
57 views

Largest four line segments of polygon

I have some polygon (see darkblue contour): It consists of very small segments, pixel by pixel, so angles differ although they seem to be the same. Visually we see 4 large line segments. How can I ...
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2answers
82 views

Prove that there does not exist a $n$-regular polygon $(n\ge 4)$, such that its sides and diagonals are all integers.

Prove that there does not exist a $n$-regular polygon $(n≥4)$, such that its sides and diagonals are all integers. Maybe a famous problem, but I don't know how to solve that.
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36 views

Minkowski sum and Polygons

The problem:.. Given two convex polygons $A$ and $B$, we can define Minkowski sum as A + B = {a + b: a $\in$ A, b $\in$ B}, where $a + b$ vector sum. Prove that: every vertex $p \in A + B$ is a ...
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1answer
60 views

Minkowski sum and vectors

Problem: Given two convex polygons A, B, we can define Minkowski sum, as A + B = {a + b: a $\in$ A, b $\in$ B}, where a + b vector sum. Prove that: for every external perpendicular u to an edge of A,...
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1answer
12 views

Degree measure of multiple polygons

I made this design on the Desmos calculator, and I was wondering what the quickest way was to find the degree measure of each individual angle. What I know so far: The measures of each of the ...
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1answer
53 views

Proof for Centroid Formula for a Polygon

I was reading a paper and I found this formula for the centroid of a polygon in terms of its coordinates but no proof was given. $C_x =\frac{1}{6A} \sum_{i=0}^{N-1}(x_i+x_{i+1})(x_iy_{i+1}-x_{i+1}...
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2answers
38 views

Area enclosed by a polygon

I did some work in the area of mensuration and came across an interesting concept/formula. The formula states that For a polygon having vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3),\dots(x_n,y_n)$, the ...
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1answer
43 views

Regular polygon inscribed in a unit circle

Given a point $P$ on the circumference of a unit circle and the vertices ${A_1},{A_2}, \ldots ,{A_n}$ of an inscribed regular polygon of $n$ side. Prove that $P{A_1}^4 + P{A_2}^4 + \cdots + P{A_n}^4$...