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
31 views

Area of minimum regular polygon given three vertices

Related question: Regular polygon determined by three vertices I have solved a problem that is related to the linked question. It boils down to the question "given three vertices of a regular ...
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2answers
20 views

Surface whose points can all be connected by straight lines contained in the surface

I don't know the mathematical term used to define a surface whose points can all be connected by staright lines contained in the surface vs cannot all be connected by straight lines contained in the ...
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0answers
17 views

Prove/Disprove Theorem about regular polygons

Given any regular polygon, and a point inside the polygon, prove that the sum of the shaded areas that formed of the point, vertices and altitudes, are equal to the some of the unshaded ones. ...
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2answers
60 views

Ways to create a quadrilateral by joining vertices of regular polygon with no common side to polygon

How many ways are there to create a quadrilateral by joining vertices of a $n$- sided regular polygon with no common side to that polygon? It's quite easy to solve for triangles for the same ...
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1answer
21 views

Sum of angles in an equilateral $n$-polygon if $n$ approaches infinity

It's obvious that in the case of an equilateral polygon, the number of angles between two sides increases in number, as are the angles themselves. Now the angle between two lines from both sides of ...
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1answer
22 views

Polygon Equal Edge Offsetting?

If I have a random polygon of any complexity, be it a square or an irregular 20 sided polygon, how can I scale this up? I know the coordinates of each point on the polygon, but that is all. Another ...
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1answer
32 views

How do I calculate the angle between two sides of a polygon? [closed]

So I got a polygon and I have all of the points. What I need, is to find all internal angles of this irregular polygon. How do I do that?
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0answers
3 views

Name for a complex but consistently wound polyline loop?

So I have an algorithn which operates on a plane region defined by a directed polyline loop. This algorithm has the unusual property of working properly for self-intersecting polylines, but only if no ...
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0answers
11 views

Non-convex subdivisions of newton polygon of a tropical plane curve

This is probably an elementary question, but how come the Newton polygon of a tropical plane curve can't have non-convex subdivisions? Or can it?
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2answers
20 views

Equilateral hexagon and a Circle

In the following diagram $ABCDEF$ is a equilateral regular hexagon with $AB = 1$ A circle is drown with radius $2$ with point $E$ as a center. What is the area of the shaded region of the circle ...
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0answers
52 views

Lost proof of trigonometric formula

The following formula seems to be true for odd positive integers $n$ but i forgot the way I proved it $$\sum_{k=1}^n\tan(\alpha+\frac{k2\pi}{n})=n\tan(n\alpha)$$ Maybe someone can deliver the ...
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1answer
28 views

What is the monotonicity of a polygon? [closed]

What is the monotonicity of a polygon and why is it necessary to check the monotonicity?
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0answers
29 views

Axis of approximate symmetry in irregular polygon

I'm searching for an axis of approximate reflection symmetry in irregular convex polygons with straight boundaries. Considering the polygons are irregular, the axis of approximate symmetry (defined as ...
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2answers
36 views

Relation between areas of regular polygons wrt constant height.

If we have a regular $n$-gon with height 1 (midpoint to furthest vertex for odd-gons/midpoints to midpoints for even-gons), how does the area of different regular $n$-gons compare to each other from ...
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0answers
14 views

The name of a polygon defined by multiple overlapping annuli

I am working on a problem in a metric space where points are partitioned into various annuli. If there exists multiple annuli that define a set of points then a polygon can be formed from their ...
2
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1answer
58 views

Prove the number of red sides are always larger than $\frac{n^{2}-2n}{2}$

Every sides and diagonals of a polygon (n-sided) are colored by red or blue. If there are no triangle that all it's sides are colored by blue, prove the number of red sides are always larger than ...
2
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1answer
26 views

The exact type of my 3d model

I have reconstructed vertical features (hole like objects lie on a vertical face) lie on two connected faces. To understand the situation, I say I have 2 walls with many windows and doors on ...
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1answer
435 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...
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1answer
49 views

Find the area of a particular pentagon associated to a given triangle

Let $ABC$ be a triangle with base $AB$. Let $D$ be the midpoint of $AB$ and $P$ be the midpoint of $CD$. Extend $AB$ in both direction. Assuming $A$ to be on the left of $B$, let $X$ be a point on ...
2
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2answers
80 views

Polygons with a Unique Triangulation

For each n > 3, find a polygon with n vertices that has a unique triangulation. I want to say that you can somehow build these polygons by continuously adding triangles somehow, but I'm not sure.
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2answers
5k views

Number of Parallel/Not Parallel Diagonals of a Regular Polygon

This is a painfully easy problem, yet the answer continues to escape me. I am seeking a general formula that can be employed to determine the number of diagonals of a regular polygon that are parallel ...
2
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0answers
39 views

Elementary proof of Jordan curve theorem for polygons

Courant described the outline of an elementary proof of the Jordan curve theorem for polygons using the order of points: The order of a point $p_0$ is defined by the net number of complete ...
3
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0answers
63 views

Existence of a polygon with specified interior angle measures

We all know that the sum of the interior angles of a polygon is $180^{\circ} (n-2)$. But is the converse true? Given a sequence of $n$ angle measures whose sum is $180^{\circ} (n-2)$, can it be ...
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1answer
33 views

How do you prove the arithmetic mean of vertices of a polygon lies in itself?

There's an n-sided convex polygon with vertices denoted by $A_1(x_1,y_1),A_2(x_2,y_2)..A_n(x_n,y_n)$. Now we draw a point $P(\frac{\sum x_i}{n},\frac{\sum y_i}{n})$, then how do you show that $P$ must ...
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7answers
770 views

Number of ways to connect sets of $k$ dots in a perfect $n$-gon

Let $Q(n,k)$ be the number of ways in which we can connect sets of $k$ vertices in a given perfect $n$-gon such that no two lines intersect at the interior of the $n$-gon and no vertex remains ...
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1answer
12 views

What's the efficient method to find the farthest vertex from centroid

Say I have a arbitrary convex polygon, what I wonder is the longest length from its centroid to its vertex, and which vertex it is. I've looked it up on Wikipedia finding that I have to calculate ...
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0answers
22 views

How to generate random boundary programmatically?

What should I read to understand how to draw programmatically random oil 'boundary' like on the picture below? Yes, it should go from the top to the bottom and so I don't need these 'long' drops ...
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1answer
301 views

Question on Proof of Shoelace Formula

I was looking for a way to prove the shoelace formula when I found this proof: For this clockwise order to make sense, you need a point O inside the polygon so that the angles form $OA_{i}A_{i+1}$ ...
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2answers
92 views

number of subset forming polygon

Given a set $S = \{ 1 , 2 , 3,\ldots, n\}$. How can I find number of subsets of size $K$ ($K < n$) whose elements taken as length of edges can form a convex polygon ($K$-sided).
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1answer
87 views

Proof involving Ramsey numbers

$S$ is a set of R(m,m;3) points in the plane in which no 3 points are collinear. I am trying to prove that $S$ contains $m$ points that form a convex $m$-gon. I have tried using similar logic to the ...
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1answer
126 views

Given the sides of a polygon, determine if it is convex or concave

We are given the lengths of all sides of a polygon. We need to determine if the given polygon is convex or concave. How can this be done? What is the propery applied to determine this?
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2answers
45 views

Prove that $\overrightarrow{GH}.\overrightarrow{IJ}=-2x^2+8x-2$ in a regular hexagon

We know that the hexagon is regular and: $\overline{AB}=1$; $\overline{AG}=\overline{CI}=\overline{DH}=\overline{FJ}=x$; How would you prove that ...
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2answers
37 views

Polygons joining together to make similar polygons

I was given the below question in a math competition a few weeks ago. I was bit confused about the wording of the problem and what was meant by the word "similar" in the given context. I tried ...
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0answers
30 views

Pentagonal tiling

I am currently working on a research project in my last year of high school. For this paper we are discussing Eschers tesselations, both in the euclidian and the non-euclidian plane. At the moment I ...
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1answer
95 views

Polygons within a polygon

r-sided polygons are formed by joining the vertices of a n-sided polygon.Find the number of polygons that can be formed,none of whose sides coincide with those of the n-sided polygon? Polygon is ...
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0answers
28 views

Movement of multiple polygons

Currently I'm trying to find a way to move an arbitrary number of arbitrary (unnested) polygons fulfilling the following constraints: The movement of all polygons is performed concurrently during ...
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2answers
17 views

Curvature measure for polygones on a 2D space

I would like to implement a curvature measure for polygones on a 2D space. My goal is to compute shape parameters to know if the polygon is close to a circle or has a sinuous shape or an elongated ...
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0answers
18 views

Best posible circumference within an irregular polygon given by linear functions?

I would appreciate any formula or theory on how to calculate the radio of the best circumference within linear functions given. Thanks in advance
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4answers
832 views

What polyhedron has 11 vertices and 17 edges

On my math test it asked me how many polygons it takes to create a polyhedron that has $11$ vertices and $17$ edges. I'd just like to see what the shape would look like and I can figure out the ...
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1answer
38 views

Find ratio of areas of triangle to pentagon?

ABCDE is a regular pentagon; rays AB and DC intersect at X. Now the area of triangle BCX is 1. What is the area of the pentagon? I figured out that the area of the pentagon is the square root of 5. ...
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1answer
26 views

Number of Pieces a regular $n$-gon is cut into by its diagonals [closed]

In how many pieces a regular n-gon is cut into by its diagonals? I need a general formula. By inspection, I have the solution to some lower values of $n$. For $n=3,4,5,6$ solutions are $1, 4, 11, ...
2
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1answer
26 views

Is there a convex polygon such that it cannot be tiled with some number of congruent connected pieces?

So the title says it all. I assume that polygons have straight line segments as their edges and that they have finite number of edges. The number $n$ of pieces is, of course, $n>1$, to avoid ...
3
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1answer
22 views

Tiling the concave polygons with non-polygons

Suppose that we reside in the set of all concave polygons (that is, polygons which are non-convex and simple, simple means that the boundary of the polygon does not cross itself). Let us denote that ...
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1answer
253 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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0answers
42 views

Minimal diagonal intersections in a convex polygon

OEIS A006561 gives the number of intersection points in the diagonals of a regular polygon. There's a paper by Poonen. For 4 vertices to 12, the number of intersection points is: $$1, 5, 13, 35, 49, ...
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1answer
42 views

Rational distance from a regular polygon.

Consider a regular n-gon with side length $A$. Let $p$ be a point in the polygon. Let the distances from $p$ to the corners of the n-gon be $x_1,x_2,...,x_n$ Are there solutions with ...
2
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1answer
31 views

How many ways can $3$ regular polygons meet at a vertex?

This is equivalent to the positive integer solutions to $$\frac{a-2}{a} + \frac{b-2}{b} + \frac{c-2}{c} = 2$$ with $3 \le a \le b \le c$. Small solutions like $(6, 6, 6)$ and $(4, 8, 8)$ can be ...
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2answers
28 views

Is it against the definition of polygon for edges or vertices to overlap or being the same point or segment

I think question shows what I'm after but I will try to add some more details. So there are two cases: I have a polygon ABCDEC'(A). ...
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0answers
18 views

Can plane be tiled with polygons

I'm working on cellular automaton, where each cell has K number of friends. For example cellular automaton with 8 friends and particular rule is Conway's Game of Life. I want to draw grid for any ...
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2answers
173 views

Computing bounding box of polytope (system of linear inequalities)

Given a N real valued variables and a set of linear inequality constraints, I would like to find a minimal bounding box which encapsulates the convex polytope defined by these constraints. I think ...