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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1answer
12 views

In every polygon circumscribed about a circle, there exist three sides that can form a triangle.

How can one show that in every polygon circumscribed about a circle, there exist three sides that can form a triangle? (This was posted by another user and then deleted while I was typing my answer.) ...
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15 views

Beautiful property of every single circunscribed polygon ever

Show that in any circumscribed polygon, there exist three sides which could form a triangle. Been on it for a while starting with quadrilaterals and trying to connect some properties and proved for ...
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0answers
6 views

create polygon section with equal sides

I have to create essentially these sections of a polygon. I have width(W) and height (H), and number of sides (3 on left abc and 4 on right image ABCD) I need each side to be equal. How can I ...
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1answer
27 views

Intersections in polygons

I'm having troubles solving the following problem which is about combinatorics: let $n$ be a natural number $\ge 3$, and a convex polygon with $n$ vertices. Each vertices are supposed to connect ...
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1answer
101 views

Largest $n$-vertex polyhedron that fits into a unit sphere

In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex ...
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1answer
20 views

How to sort vertices of a polygon in counter clockwise order?

How to sort vertices of a polygon in counter clockwise order? I want to create a function (algorithm) which compares two vectors $\vec v$ and $\vec u$ which are vertices in a polygon. It should ...
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0answers
15 views

Permutation of conjoined faces in regular polygon with diagonals

I've been doing some study on relationships in polygons, right now, regular polygons. I've been trying to find relationships between the diagonals, angles, faces, vertices, and primarily conjoined ...
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12 views

$C^{\infty}$ distance function for polygons?

I'm exploring some novel/weird ways of doing continuous collision detection. Suppose I have two (possibly concave) polygons in $\mathbb{R}^2$ and I want to find the "distance" between them at a ...
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33 views

Factorization of seventh cyclotomic polynomial

The fifth cyclotomic polynomial $\Phi_5(z)$ factors as $$ \Phi_5(z)=(z^2+\varphi z+1)(z^2+(1-\varphi)z+1) $$ where $\varphi$ and $1-\varphi$ are the solutions to $x^2-x-1$. Of course, $\varphi$ is ...
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1answer
40 views

Parallelograms in a convex hexagon

If you have a (convex) hexagon, and label it ABCDEF, and if ACDF is a parallelogram, and ABDE is a parallelogram, prove that BCEF is a parallelogram also.
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1answer
16 views

Properties of area of simple polygon with integer coordinates

Is the area of a simple polygon with integer coordinates is half an integer? I was reading shoelace formula and this occurred to me.
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1answer
14 views

Is there any polygon exists such that sum of length of square of two adjacent sides is equal to another side/diagonal?

In Right angle triangle we have $ a^2 + b^2 = c^2$ where $a^2 = (x_1-x_2)^2 + (y_1-y_2)^2 ,$ $b^2 = (x_3-x_2)^2 + (y_3-y_2)^2 $and $c^2 = (x_1-x_3)^2 + (y_1-y_3)^2$ And in Square we have $ a^2 + ...
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1answer
22 views

Arithemic progression

All the interior angles of the polygon form A.P. The common difference is 6 degree..the greatest angle is 135 degree..find the number of sides of polygon. I try to solve by using ...
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2answers
192 views

How to find the number of squares formed by given lattice points?

Let us say that we are N integer coordinates (x, y) - what would our approach be if we were supposed to find the number of squares we could make from those given n points? Additionally, if we were to ...
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1answer
118 views

Count ways to form isosceles triangles

Their are N persons sitting on a table with N vertices.We need to count the number of isosceles triangles formed such that each vertex of the triangle is a vertex of the table and all persons seating ...
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1answer
11 views

A polygon compactness metric that filters out the noise of small concavities?

I am trying to characterize the compactness of polygons. I have come across this definition of "compactness ratio" (perimeter/area), which works for most polygons, but I find as the negative concave ...
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0answers
39 views

Triangulations of the concave polygon

It is known that the amount of possible triangulations of the convex polygon by disjoint diagonals is the Catalan number. But can we somehow know possible amount of the triangulations of the concave ...
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0answers
64 views

Is there a 4 pointed star that is regular?

I am studying about the area of a 4 pointed star, I wonder if there is really a 4 pointed star that is regular? what could be the characteristics of a regular 4 pointed star? that is how i ...
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1answer
254 views

Convex n- sided polygon proof writing (homework question)

Would anyone be able to help me with the following problem or give me a push in the right direction? I am not entirely sure where to start and I have been looking at this problem for hours... Any help ...
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4answers
197 views

Pentagon Geometry

$ABCDG$ is a pentagon, such that $\overline{AB} \parallel \overline{GD}$ and $\overline{AB}=2\overline{GD}$. Also, $\overline{AG} \parallel \overline{BC}$ and $\overline{AG}=3\overline{BC}$. ...
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0answers
13 views

Skipping points on a polygon

Suppose I have an N sided regular polygon with vertices labelled 1 to N. If I do (1,2,3,4,...,N), I trace the perimeter of the polygon. If I do (1,3,5,...,N) I will form a star shape and end up back ...
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4answers
66 views

Can anyone give me x,y coordinates for an octagon?

I am looking to draw a octagon and I need $(x, y)$ coordinates.
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0answers
43 views

Can a polygon with minimal perimeter self-intersect?

Recipe. Do the following. Throw $N$ random points $(x_0,y_0),(x_1,y_1),x_2,y_2),\cdots,(x_{N-1},y_{N-1})$ in the plane.Define $(x_N,y_N)=(x_0,y_0)$ : enumeration is $\mod N$ . These points are joined ...
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0answers
17 views

Polygons inscribed in circles, with integer sides and integer radius

Is there a simple characterization for an integer partition $(s_1,\dots,s_k)$, such that a polygon with these sides is inscribed in a circle with integer radius? This is what I got so far: All ...
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1answer
29 views

Selecting a random orthogonal polygon

For a certain demo application, I want to create at random a rectilinear polygon with a given number of corners. Selecting random $x$ and $y$ coordinates of each corner is not a good method, since ...
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0answers
24 views

Looking for algorithm for spherical point in polygon that works across meridian and anti-meridian

I need to process millions of latitude/longitude points every day to see if they are located within a defined lat/lon bounded polygon. The polygon may be rectangular, or it may be some irregular 3.. ...
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1answer
25 views

Area Of Polygon Whose Edges Are In Given Distance From A Given Polygon Edges

I'm handling a problem which I find quite difficult to solve; My input is a changing number of coordinates (real GPS coordinates), usually I get 4-8 coordinates, and another number,which indicates a ...
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0answers
43 views

Find circles that completely cover a polygon minimizing the amount of space covered outside the polygon

I have an arbitrary polygon that I need to roughly represent using circles. Any point inside the polygon must lie inside a circle. There will be points outside the polygon that will fall under a ...
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0answers
13 views

For an app teaching about polyhedra, what are some core characteristics to include?

For fun: I'm building a 3d app that teaches about polyhedra. What should I include? The obvious didactic elements for each polyhedron would be: Fundamental polygon's Vertices 
Edges
 Faces
 (and ...
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3answers
32 views

Find the measure of angle E.

http://static.k12.com/eli/bb/811/7537/0/2_36640_44211/7537/cfcbab7622b25115e3996826ebe54350776a6601/media/a0fb44a9ac3761c0d89bd1c3ffa513c508eb78bf/mediaasset_650483_1.gif help please i still mix the ...
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0answers
31 views

Volume of a rotated regular polygon

I want to calculate the volume of the shape which is created when you rotate a regular $n$-sided polygon around the $y$ axis with a major radius $r$. (like a torus, but with a polygon as rotated ...
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1answer
44 views

Cut the Cake into 4 parts

I'm facing the following problem: I'm given a set of coordinates on an integer grid that define the vertices of a polygon. The polygon is guaranteed to be convex. It's proven that such a polygon can ...
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0answers
17 views

Jordan Curve Theorem Coordinate Order

So I am trying to wrap my head around the Jordan curve theorem and can't seem to understand why the order of the vertices doesn't matter. Why do the coordinates not have to be presented in either a ...
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1answer
39 views

splitting trapezoid

I have a trapezoid. I know it's height and bases. How can I split it in 2 parts of given area by line parallel to bases. For example my trapezoid has area of S. And I want to get 2 trapezoids with ...
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2answers
91 views

splitting polygon in 4 equal parts

I have a convex polygon and I want to divide into 4 equal parts using the two perpedicular splits. Like in a picture. I need s1 = s2 = s3 = s4; I need to get coordinates of point where the lines ...
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1answer
35 views

What's wrong with this pseudocode for Forster-Overfelt's version of the Greiner-Horman polygon clipping algorithm?

The Problem I'm trying to understand and implement the Forster-Overfelt version of the Greiner-Horman polygon clipping algorithm. I've read the other Stackoverflow post about clarifying this ...
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4answers
66 views

Smallest and largest possible angles of given polygon

What is the smallest and largest possible angle of a triangle? (my guess = 1, 178) What is the smallest and largest possible angle of a quadrilateral (convex or concave doesn't matter, and also ...
3
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1answer
31 views

finding parallel sides from a irregular decagon?

Is it possible to find out that which of two sides are parallel in this irregular decagon.If,it is yes;then how can I proceed? I have tried with "Consecutive Interior Angles".but can't come to a ...
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0answers
17 views

How do I write my reference for a polygon

Given a regular pentagon ABCDE. Angle EAB = x + y I have this formula: x + y = (1/5)(180)(5-2) But how do I write the ...
3
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2answers
38 views

What is the ratio of the side length of a regular hepatgon to the side length of the internal heptagon?

Given a regular heptagon with side length 1, create a star heptagon by connecting every vertice. Note that removing the "points" of the star yields a similar heptagon. I want to know the side ...
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1answer
21 views

recutting a polygon to obtain its flip

I am curious if it is possible to cut the triangle with vertices (0,0), (0,1), (1,0) into many polygonal pieces and only translate each piece appropriately to construct the triangle with vertices ...
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2answers
54 views

Derivation of the formula for the area of a regular polygon given the side length.

There are many formulas for finding the area of a regular polygon- but this is the one I am interested in: $$A=\frac{S^2n}{4\tan(\frac{\pi}{n})}$$ where $n$ is the number of sides, and $S$ is the ...
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1answer
41 views

Given a polygon of n-sides, why does the regular one (i.e. all sides equal) enclose the greatest area given a constant perimeter.

This doesn't require much more than the title. I just need an explanation, but an algebraic proof would be a bonus. We can demonstrate this for quadrilaterals, a square is best as shown by this ...
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3answers
59 views

Prove that $n\sin\frac{2\pi}{n}-\frac{1}{4}n\sin\frac{4\pi}{n}>\pi$ (corrected inequation)

Prove that Prove that $n\sin\frac{2\pi}{n}-\frac{1}{4}n\sin\frac{4\pi}{n}>\pi$ algebraically or geometrically. $n\sin\frac{2\pi}{n}-n\sin\frac{\pi}{n}$ means the area of a regular n-gon + the area ...
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3answers
54 views

Where is the interior of the polygon?

There is an axis-parallel (orthogonal) simply-connected polygon given as a list of corners. How can I know whether a certain vertical segment has the interior of the polygon on its east or on its ...
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2answers
54 views

Maximum number of vertices in intersection of triangle with box

Suppose we have a triangle and a box in 3D. The intersection of the triangle with the (solid) box will be a polygon with some number of vertices (possibly zero). The vertex count will vary according ...
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3answers
269 views

Area of the Limiting Polygon

Start with an equilateral triangle with unit area. Trisect each of the sides and then cut-off the corners. In this case, we get a regular hexagon - see the picture below. Next, trisect each of the ...
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0answers
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Best convex bounding polygon from a set of given lines

Given a polygon $P$ and a set of predefined lines, I am looking for the subset of lines that creates the best fitting convex polygon with respect to $P$. In other words the area of an ...
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0answers
16 views

Shortest path over polygons with different cost of travel

The real world problem I'm trying to solve: A road has to be built over a terrain. The terrain have areas that are more costly to build over. Those areas can have complex shapes. How do I build the ...
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1answer
34 views

Function of polygonplots does not plot all polygons

Could anyone explain to me why my Maple-Code does not work properly? ...