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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Why reflection and rotation are sufficient operations in dihedral group?

I know a bit of elementary group theory but please ignore dihedral group in the title and let's make it simple enough so a high student can read the question and the answer(s)... Suppose we have a ...
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1answer
48 views

Why the number of symmetry lines is equal to the number of sides/vertices of a regular polygon?

Considering the/a definition of a regular polygon from Wiki : In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all ...
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77 views

Placing circles inside of a regular polygon.

Alice and Bob play the following game: on a table there is a regular $n$-gon. On each person's turn, they are required to place a circle of radius $r$ fully in the interior of the $n$-gon such that it ...
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38 views

There exist three consecutive vertices A, B, C in every convex n-gon with n≥3, such that the circumcircle of triangle ABC covers the whole n-gon

From Problem Solving Strategies by Arthur Engel: Problem to prove: There exist three consecutive vertices $A$, $B$, $C$ in every convex $n$-gon with $n \ge 3$, such that the circumcircle of triangle ...
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Area covered by fixed perimeter around polygon.

Suppose I have a polygonal field with a post at each vertex and a non-extensible rope threaded through each post around the perimeter but with some slack. How can I determine the perimeter of the area ...
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49 views

Centroid and circumcenter — how close?

Suppose $R$ is some planar region, bounded by a curve. Let $C_1$ be the centroid of $R$, and let $C_2$ be the center of the "circumcircle" (the smallest circle enclosing $R$). Intuitively, it seems ...
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70 views

Calculating radius of circles which are a product of Circle Intersections using Polygons

Lets say you imagine a circle with the radius $R$ and you inscribe a regular polygon with $n$ sides in it, whose side we know will then be: $$a=2R*sin(\frac{180}{n})$$ Then you draw a set of circles ...
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consider the following statements regarding the smallest interior angle of a n sided polygon with perimeter n units and with maximum area?

let(f) be the relation defined by f(n) = The smallest interior angle value of the n sided polygon with perimeter n units with maximum area, for each positive integer n(>2).which of the following are ...
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40 views

Why am I getting the wrong formula for the area of a dodecagon?

More likely than not, I'm just making a simple algebraic mistake, but I can't seem to find it and so I would like some help. Divide a (regular) dodecagon into $12$ congruent isosceles triangles with ...
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1answer
63 views

Interior angles of irregular quadrilateral with 1 known angle

I have the measurements of the four sides of an irregular polygon and I need to find out the size of each interior angle. I know the sum of the angles is 360 degrees but because it's not a regular ...
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3answers
37 views

Triangles with no common side in a polygon

There are n sides of a polygon(where $n>5$). Triangles are formed by joining the vertices of the polygon. How many triangles can be constructed with no side common to the polygon? My try: Total ...
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Is there a theoretical (or practical) definition of $n$-gon, for $n < 0$?

Background This is purely a "sate my curiosity" type question. I was thinking of building a piece of software for calculating missing properties of 2D geometric shapes given certain other properties,...
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Number of sides of regular polygon

What I need to find: Number of sides of a regular polygon What I am given: Any 3 vertices of the polygon What I currently know: I can find the center of the polygon. That would be the intersection ...
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1answer
44 views

Does the inner pentagon inside a Robbins pentagon $also$ have a rational area?

The Heron triangle has integer sides and area. The Robbins pentagon is just the generalization: it also has integer sides and area. The example below has sides $78, 126, 66, 50, 32$ and area $A_R = \...
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30 views

polygon angle names

Regular polygons can be divided up into triangles of equal sizes. For example, a pentagon would have 5 triangles. Each would have one angle with a value of 72 degrees and two angles of 54 degrees. ...
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What is circumradius $R$ of the great disnub dirhombidodecahedron, or Skilling's figure?

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$). For irrational ...
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1answer
114 views

Finding $\sqrt{17}$ and $\sqrt{257}$ in the regular $17$-gon and $257$-gon?

(Edit: I need to revise this question with my original intent. Pls do not answer it yet. Thanks.) Given the regular $n$-gon formed by the $n$-th roots of unity. For some $n$, how do we find $\sqrt{...
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40 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 polygon,...
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2answers
24 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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2answers
84 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
30 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
32 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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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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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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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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23 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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31 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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43 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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48 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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18 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 ...
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1answer
61 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 $\...
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53 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 $...
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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
39 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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1answer
17 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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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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2answers
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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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215 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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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 $\overrightarrow{GH}.\overrightarrow{IJ}=-2x^2+8x-...
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1answer
94 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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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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1answer
51 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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147 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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29 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
24 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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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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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
29 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, ...
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1answer
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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 ...
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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 ...