All Questions
Tagged with polygons euclidean-geometry
158
questions
3
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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
votes
0
answers
47
views
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
views
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
votes
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
0
votes
1
answer
110
views
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
votes
0
answers
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
views
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
views
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
views
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
0
votes
0
answers
34
views
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
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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