All Questions
Tagged with polygons euclidean-geometry
156
questions
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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
285
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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
253
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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 ...
- 2,212
7
votes
2
answers
238
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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
0
votes
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
-1
votes
3
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86
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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
234
views
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
0
votes
0
answers
69
views
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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votes
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58
views
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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129
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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
votes
0
answers
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:...
- 31
0
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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 ...
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