All Questions
Tagged with polygons euclidean-geometry
32
questions with no upvoted or accepted answers
8
votes
0
answers
146
views
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
7
votes
0
answers
209
views
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
7
votes
0
answers
619
views
Polygons with coincident area and perimeter centroids
Let $P$ be a simple, planar polygon.
Define $c_a$ as the area centroid of $P$, i.e., the center of gravity of the
closed shape $P$.
Define $c_p$ as the perimeter centroid of $P$,
the center of gravity ...
- 30.4k
6
votes
0
answers
312
views
Construction involving regular polygons inside a circle
Let's make a construction involving regular polygons:
► First, we begin with a equilateral triangle, with side $\ell_3 = 1;$
► After, we draw a square on the middle point each side of the initial ...
- 635
3
votes
0
answers
197
views
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
3
votes
0
answers
63
views
Change of angle inside a quirky hexagon
So I am dealing with the hexagon as shown in the picture below and I need to find out how one angle depends on another angle. More specifically, I need $\frac{d\psi}{d\varphi}$ at $\varphi=0$.
Note ...
- 157
3
votes
0
answers
476
views
Better "centerpoint" than centroid for placing a map marker inside a concave region (that may have holes)?
I'm using the centroid of polygons to attach a marker in a map application. This works definitely fine for convex polygons and quite good for many concave polygons.
However, some polygons (banana, ...
- 141
3
votes
0
answers
125
views
A convex $n$-gon and the $n$-gon made by its $n$ medians
For a convex $n$-gon $P_1P_2\cdots P_n$, let $M_i$ be the mid-point of the line segment $P_iP_{i+1}\ (i=1,2,\cdots,n)$ where $P_{n+1}=P_1$. Also, let $Q_1Q_2\cdots Q_n$ be an inner $n$-gon made by $n$ ...
- 145k
2
votes
0
answers
193
views
Maximum distance between a point and a convex polygon as a function of its perimeter
Let $P$ be a convex polygon with perimeter $N$, and let $x$ be an arbitrary point. Show that the maximum distance between $x$ and a vertex of $P$ is at least proportional to $N$ (i.e. there exists a ...
2
votes
0
answers
220
views
Reconstructing a polygon from the Midpoints of Its Sides
I was reading through Dijkstra's 'A Collection of Beautiful Proofs' and stumbled upon this elegant piece of work:
11. Reconstructing an odd polygon from the midpoints of its sides.
We shall ...
- 21
2
votes
0
answers
89
views
Area enclosed by locus of points at fixed max distance from a closed curve
There is a question that goes as follows:
The locus of $P$ is the interior of an n-sided convex polygon of
perimeter $p$, area $\Delta$. The locus of $Q$ is all points where
$PQ\leq r$. Find ...
- 2,032
1
vote
0
answers
19
views
Intersection of Symmetric Convex Sets
Given a symmetric convex compact set $K\in\mathbb{R}^2$, show there are no sequences $(r_i)_{i=1}^n$ of positive scalars and $(P_i)_{i=1}^n$ of points in $\partial K$, so that $n>1$ and
$$\sum_{i=1}...
- 630
1
vote
0
answers
73
views
How to constrain a rectangle within an arbitrary 2d polgyon?
I am working on a graphics project wherein I need to keep a rectangle (assume it has a fixed but arbitrary size, rectWidth * rectHeight) constrained inside an arbitrary polygon.
The polygon is ...
- 153
1
vote
1
answer
57
views
non-convex even-sided polygons whose vertices lie on a circle
Given $2n$ evenly spaced points on a circle, the opposite sides in the convex polygon (formed by these points) are parallel. If we remove the requirement of convexity, I get degenerate polygons that ...
- 400
1
vote
1
answer
101
views
Proving equal angle in regular pentagon
showing that all the angles in a regular pentagon are the same euclid style.
In this other page, How can one of the answer get to the conclusion of
BED + ABE = ABC
This is my first time using, ...
- 43