All Questions
Tagged with polygons euclidean-geometry
158
questions
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25
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Rational quantities associated with a bicentric heptagon
For odd reasons of my own, I would like to create a math puzzle involving a bicentric heptagon (i.e., a 7-sided polygon which has both an inscribed and a circumscribed circle). The puzzle is to be ...
- 547
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0
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26
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Polygon Boundary in 3D Space
A simple polygon (interior included) is a manifold with boundary being homeomorphic to a closed disk. Its (manifold) boundary is a non-intersecting closed polygonal chain. Viewed as a subset of the ...
0
votes
1
answer
57
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Finding the Areas of Polygons from Side Lengths
I am aware of the formula for the area of a regular polygon:
$A=([Side Count] \times [Side Length] \times [Apothem Length])/2$
However, I could not find an equation for the area of a non-regular ...
13
votes
4
answers
1k
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The sum of the squares of the diagonals in a polygon
The first question that got me here:
A regular dodecagon $P_1 P_2 P_3 \dotsb P_{12}$ is inscribed in a circle with radius $1.$ Compute$
{P_1 P_2}^2 + {P_1 P_3}^2 + {P_1 P_4}^2 + \dots + {P_{10} P_{11}}...
- 229
3
votes
1
answer
106
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Olympiad Trapezoid Problem about Lengths
I'd like some help with the following Olympiad Problem about a trapezoid:
There is a trapezoid $ABCD$ with parallel sides $BC$ and $AD$ such that $AB=1$, $BC=1$, $CD=1$ and $DA=2$. Let $M$ be the ...
- 467
2
votes
1
answer
290
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Constructing bicentric pentagon
I'm trying to construct a bicentric pentagon in geogebra. I read on Wikipedia that a Pentagon is bicentric if and only if it satisfies this formula $$r(R-x)=(R+x)\left(\sqrt{(R-r)^2-x^2}+\sqrt{2R(R-r-...
- 4,196
1
vote
1
answer
69
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Getting a point in the interior of a polygon without relying on winding order?
I am given an arbitrary set of points embedded in 3D.
The points are guaranteed to be ordered such that their order yields a simple closed polygon, but there is no information about whether they wind ...
- 3,439
5
votes
1
answer
258
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What is a gyrational square in this context?
This is a diagram of quadrilaterals and their duals. Within this diagram, there is a square (shown with 8 lines of symmetry) and right below that there is a "gyrational square" shown with no ...
- 2,250
7
votes
2
answers
241
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How to enumerate unique lattice polygons for a given area using Pick's Theorem?
Pick's Theorem
Suppose that a polygon has integer coordinates for all of its vertices. Let $i$ be the number of integer points interior to the polygon, and let $b$ be the number of integer points on ...
- 1,913
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3
answers
101
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Construction of a pentagon in which an angle bisector at a certain vertex is also a perpendicular bisector of a side opposite to that vertex
Construct a pentagon $ABCDE$ if lengths of its sides are $a,b,c,d$ and $e$ ($|AB|=a,|BC|=b,...$), and the bisector of the angle at vertex $D$ is also the perpendicular bisector of $AB$.
I know that $...
- 429
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3
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length of side of a regular $n$-gon is less than length of any diagonal
In any regular polygon with $n\ge 4$ sides, why is any side length strictly length to any diagonal length? (A diagonal is defined as the line segment joining non-adjacent vertices)
This is intutitvely ...
user10575
4
votes
2
answers
235
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Trajectory of light rays in a mirror polygon
Given a general polygon and we are given a ray of light bouncing between the sides of the polygon where each side is a mirror. they hit at points $P_1,P_2...$, we define $\alpha_i$ to be the smaller ...
- 2,225
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69
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Number of edges vs number of vertices in $\mathbb{R}^2$?
I was thinking about the name "Triangle", when I realized that although we usually think of polygons in terms of the number of their sides. However, when I searched the origin of the word &...
- 17
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59
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Every triangulation of a simple closed polygon in the plane has a shelling
This is exercise 2 from Ch. 1 of "Computational Topology: An Introduction" by Edelsbrunner & Harer:
Consider a triangulation of a simple closed polygon in the plane, but one that may ...
- 2,454
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133
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Finding orientation of a rectangle using the points sampled on its surface
If I have $n$ number of vectors $p \in \mathbb{R}^2$ on a surface of a rectangle, would it be possible to estimate the underlying rectangle orientation? The rectangle show in the figure is a virtual ...
3
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197
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Regular heptagon coordinates from a cubic field
Find coordinates for a regular heptagon in 3D Euclidean space where all $3$ components $(x,y,z)$ of all $7$ coordinates are elements of the same cubic field, or prove that it can't be done.
Background:...
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47
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Combining polygons to make a similar one
One can combine two identical rectangles to form a larger rectangle which is a scaled version of the smaller pieces -- this is the case for two A5 pages combining to A4 page.
Are there any other ...
- 1,822
5
votes
1
answer
94
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If a cyclic polygon with at least four sides has an incircle, must it be regular?
Suppose that the vertices of a polygon with four or more sides lie on a circle, and that another (possibly non-concentric) circle touches each of its sides. Intuitively, it seems to me that the ...
- 19k
4
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2
answers
189
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Does any edge-to-edge tiling of the Euclidean plane by convex regular polygons have only demiregular vertex configurations?
In the Euclidean plane, a vertex figure of an edge-to-edge tiling by convex regular polygons is called demiregular if and only if its vertex configuration is $3.3.4.12, 3.3.6.6, 3.4.3.12,$ or $3.4.4.6$...
- 4,057
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votes
1
answer
110
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Can a non-degenerate polygon with all sides equal have unequal angles?
I have always been hearing that a regular polygon is a polygon with equal sides and equal angles, but I never considered the fact that it may be possible for a polygon with all sides equal but unequal ...
8
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146
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Which objects can be Minkowski halved?
The Minkowski Sum of two subsets $A,B \subset \mathbb{R}^n$ is
$$A \oplus B = \{a + b | a \in A, b \in B\}$$
For a given $A$, is there some condition that tells me when I can find a $B$ such that $A = ...
- 2,339
8
votes
1
answer
150
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How to find the $2d$ cross-sections of the $n$-hypercube that are regular $2n$-gons?
I'm currently trying to do something that sounds doable, but on which I've now been stuck for a while.
The idea is that, if you consider an $n$-hypercube, the polygon found by doing a $2d$ cross-...
- 83
4
votes
1
answer
170
views
Ratio between circumference and "radius" of a polygon
Given some polygon $P$ in two-dimensional Euclidean space, I want to define the radius of $P$ as the average of the radii of the smallest outer circle and the largest inner circle. An outer circle has ...
- 73
2
votes
1
answer
239
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Find the length of $HI$ in the regular heptagon
I found the problem below in Twitter
$ABCDEFG$ is a regular heptagon. $EFHD$ is a rhombus and $HI$ is drawn perpendicular to side $AB$ find the length $HI(d)$
What I've done so far:
I can find the ...
- 773
0
votes
1
answer
283
views
Geometric inequality in regular pentagon
Let $ABCDE$ a regular pentagon inscribed in a circle of center $O$. Let $P$ an interior point of the pentagon from which we consider parallel line segments to all the sides of the pentagon. We know ...
9
votes
3
answers
399
views
Show that U,V and H are colinear
We are given a regular icosagon as below:
I wanna prove that the red line exists.
I know that $U$ is the incenter of $\triangle TLB$ ($T,U,G$ are collinear)
I know that $V$ is the incenter of $\...
- 1,940
3
votes
1
answer
208
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Find the area of a regular pentagon as a function of its diagonal
For reference:
Calculate the area of a regular pentagon
as a function of its diagonal of length $a$. (Answer:$\frac{a^2}{4}\sqrt\frac{25-5\sqrt5}{2}$)
My progress:
$R$ = radius inscribed circle
$...
- 7,031
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0
answers
34
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How much of the surface of these equilateral triangles would be lit?
Consider an equilateral triangle $\Delta ABC$ in 3D with $A=(1,0,0)$, $B=(0,1,0)$ and $C=(0,0,1)$ as well as its mirror image $\Delta A'B'C'$ with $A'=-A$, $B'=-B$ and $C'=-C$. We assume that these ...
- 2,646
4
votes
1
answer
114
views
Prove that in a $4n$-gon, every other diagonal passes through a common point
Suppose two regular $2n$-gons in the plane, which interesect one another to form a $4n$-gon. Prove that every other diagonal of this $4n$-gon, i.e. $P_{1}P_{2n+1},P_{3}P_{2n+3},...,P_{2n-1}P_{4n-1}$ ...
- 247
1
vote
1
answer
86
views
Incircle of polygon tangent to a point
How can we find the largest incircle (not sure if it is still called incircle) tangent to a given point on a side of a polygon? Instead of being tangent to all sides of the polygon, it will be tangent ...
- 288
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0
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165
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2D packing problem - how to optimise/maximise area of a a set of irregular convex polygons within a polygon?
I'm interested in a particular case of this problem: fitting odd-shaped polygons/shapes within the bounds of a rectangle.
Say you have 10 sticker designs and you want to fit them all on a sheet of A4 ...
- 151
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votes
2
answers
347
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Non-trig solution for AMC 10 Question
Equiangular hexagon $ABCDEF$ has side lengths $AB=CD=EF=1$ and $BC=DE=FA=r$. The area of $\triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $r$? AMC 10A ...
- 1,834
3
votes
3
answers
215
views
Find Interior Angles of Irregular Symmetrical Polygon
Apologies if this is has an obvious answer, but I've been stuck on this for a bit now.
I've been trying to figure out how to make a symmetrical polygon with a base of m length, with n additional sides ...
- 31
7
votes
0
answers
209
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Inequality conjecture for convex pentagons
Let $X_1, ..., X_5$ be the vertices of a convex pentagon with perimeter $p$ with its centroid at the origin, satisfying $d(X_i, X_j) < \frac{p}{3}$ for $1 \leq i,j \leq 5$, where $d$ is the ...
- 1,460
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1
answer
470
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Irregular convex quadrilateral: Find the diagonal length given all sides and one angle
I stumbled into an Irregular convex quadrilateral in one of my projects and I cannot figure out if the diagonal |BD| can be found.
Problem:
I have an Irregular convex quadrilateral ABCD defined by the ...
- 3
4
votes
2
answers
397
views
If all the sides of an n sided polygon are equal. Is it always a regular polygon?
For an $n$-sided polygon, if all the sides are equal, is it a regular polygon?
If yes then why is it defined to have equal angles?
If not so, how to prove that all angles are equal?
edit: I meant it ...
- 1,064
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3
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219
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Determine the side length of the regular hexagon which an arbitrary point lies on
We've found ourselves having to solve this peculiar problem for a plastic part that we are machining. I'll spare you the details.
Given an arbitrary point $P = (x, y)$ on the cartesian plane, and &...
- 418
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0
answers
63
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Finding a hexagonal construction with parallel lines
This is related to my previous question here: Finding a way to describe the position of vertices of $n$ nested hexagons
In the setting described in my previous question, I wanted to find a set of ...
user931598
2
votes
1
answer
115
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Finding a way to describe the position of vertices of $n$ nested hexagons
I am trying to find a way to describe precisely the points of $n$ nested regular hexagons when they are in the same position i.e. essentially there are $3$ lines (diagonals) with the total of $2n$ ...
user931598
5
votes
2
answers
169
views
Must a $𝐶^1$ curve with constant angular momentum alternate between a straight line or a circular arc?
Let $\alpha:(0,L) \to \mathbb{R}^2$ be a $C^1$ curve satisfying $|\dot \alpha|=1$, and assume that $\alpha(t) \times \dot \alpha(t)$ is constant.
Does one of the following hold?
$(1)$ $\alpha$ is ...
- 25.3k
4
votes
2
answers
84
views
A polygon with constant angular momentum bounds a circle
Let $\alpha:[0,L] \to \mathbb{R}^2$ be a piecewise affine map satisfying $\alpha(0)= \alpha(L)$ and $|\dot \alpha|=1$. Supopse that $\alpha(t) \times \dot \alpha(t)$ is constant.
How to prove that $\...
- 25.3k
2
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1
answer
143
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Given a quadrilateral with 4 equal areas, prove that it is a parallelogram
I have the next quadrilateral with midpoints E F G H. The source of the problem is my class of geometry, I read the book but I don't find anything related to this.
I found in Google about Varignon's ...
1
vote
2
answers
102
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Are these simple statements about polygons true?
Let $P$ be an $n$-gon with $n \gt 3$. I'm looking for proofs or counterexamples for the following statements:
there exist consecutive vertices $A,B,C$ of $P$ such that $\triangle ABC \cap \partial P =...
- 295
5
votes
2
answers
304
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Ratio of area of triangle and hexagon
I am looking for the proof of the following claim:
Claim. If the sides of the triangle are partitioned into $n$ equal segments for $n$ an even integer and each division point adjacent to the ...
- 12.9k
1
vote
1
answer
568
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Is there any bijective mapping between a non-convex polygon and a convex polygon?
Given an arbitrary non-convex polygon (2d) with $n$ vertices, is there any procedure by which to map it to a single convex polygon, and then back again? Maybe operating somehow on the angles between ...
- 1,454
3
votes
2
answers
355
views
Collinearity in bicentric pentagon
Can you provide a proof for the following claim:
Claim. The circumcenter, the incenter, and the excenter of the pentagon formed by diagonals in a bicentric pentagon are collinear.
GeoGebra applet ...
- 12.9k
1
vote
1
answer
905
views
Constructing an isosceles trapezoid with a specific decomposition into triangles
A recent question asked about finding the ratio of the bases for the following isosceles trapezoid:
That problem has been solved, obtaining a result of $|CD|/|AB|=1-1/\sqrt{2}$. What I'm curious how ...
- 16.8k
0
votes
2
answers
117
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Excircle and parallelogram
Can you provide a proof for the following claim:
Claim. Given any parallelogram $ABCD$ and excircle of triangle $\triangle ABC$ oposite to vertex $A$. An arbitrary tangent $t$ is constructed to the ...
- 12.9k
2
votes
1
answer
167
views
An ellipse determined by circumcenters
Can you prove or disprove the following claim:
Claim. A convex hexagon $ABCDEF$ is circumscribed about an ellipse. Let $G$ be the point of concurrency of hexagon's principal diagonals , and let the ...
- 12.9k
-2
votes
1
answer
58
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How to calculate the number of side of a polygon?
I stuck on this problem. Please suggest any hint on how to solve this problem. Thanks in advance.
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