# Tagged Questions

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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### Counting Regular polygons in Complete Graphs

The figure shows the correct $24-$gon, which held all the diagonals. a) Find out how we got right triangles and squares (question for arbitrary $n$)? b) How this problem can be generalized (if it is ...
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### Value Of $\pi$ obtained using limits!

What i thought was simple, a circle can be formed by increasing the number of sides of regular polygon( like pentagons, hexagons, etc ) up to infinity by keeping the distance between the center and ...
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### Predicting Spirals

I am currently in the process of analyzing a polyspiral, a spiral where each successive length drawn is increased at specified increment at the same angle. *Please note the angles selected are the ...
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### Calculating radius of circles which are a product of Tangent Intersections using a Regular Polygon

Introduction Lets have a regular polygon of $n$ sides inscribed in a circle of radius $H$, then construct tangents between the circle and each point of the polygon and draw new circle(s) trough the ...
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### For the parallelogram, Prove $XY=CD$

$ABCD$ is a parallelogram. The bisectors of $\angle A$ and $\angle B$ meet BC and AD at X and Y respectively. Prove that $XY=CD$? Please give me some hint to prove it. I can't initiate the problem so ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...