**0**

votes

**0**answers

48 views

### A generalization of the Tucker circle theorem and the Thomsen theorem associated with a conic

I gave a generalization of the Tucker circle theorem and the Thomsen theorem at here. Now, I give a more generalization of these theorems as following:
Problem: Let $A_1A_2A_3A_4A_5A_6$ be a hexagon, ...

**45**

votes

**12**answers

55k views

### If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What ...

**4**

votes

**1**answer

633 views

### Explicitly describing the region of the plane “outward of” a simple, open, oriented, cubic curve $c:(0,1)\to\mathbb{R}^2$

Some Context:
I'm working with some data given in the form of Bezier curves. I need to sort these (partially ordered) Bezier curves by "outwardness" (described below) and have come across an ...

**3**

votes

**1**answer

247 views

### Does the collection of algebraic/number-theoretic methods applied to Euclidean Geometry have a name?

I am currently writing an essay on the history of geometry. To educate myself on the subject, I sometimes read the following Wikipedia article on the history of Euclidean Geometry. It seems to me that,...

**10**

votes

**0**answers

745 views

### Dao's theorem on six circumcenters associated with a cyclic hexagon

This questions from Ngo Quang Duong's paper
In 2013, O. T. Dao published without proof a theorem with title Another seven circles theorem in Cut the Knot, a free site for popular expositionsof many ...

**9**

votes

**2**answers

234 views

### Generalization of Stewart's theorem?

I'm curious about the generalization of Stewart's theorem to more dimensions. MathWorld mentions that there is a generalization done by Bottema, but I could not find much information on it. All I ...

**10**

votes

**1**answer

388 views

### The geometric median of a solid triangle

Let $\Omega\subset \mathbb R^n$ be a compact subset of dimension $n$. Define the geometric median on $\Omega$ as the point $m_{\Omega}\in \mathbb R^n$ such that the integral $\int_{\Omega}|x-m_{\Omega}...

**4**

votes

**0**answers

151 views

### Optimal instructions for the modular construction of rectlinear Lego structures

Let $X$ be a compact (or periodic) union of integer translates of unit cubes such that the interior of $X$ is connected. (If it makes any difference, suppose that the dimension $n$ of $X$ is 3.) I am ...

**0**

votes

**4**answers

608 views

### Characterization of Angles Trisectable with Straightedge and Compass

Lindemann's prove of the transcendence of $\pi$ has settled the question, whether an arbitrary angle can be trisected, using straightedge and compass alone, to the negative.
In the following, ...

**6**

votes

**1**answer

198 views

### Law of sines for tetrahedra

Wikipedia gives a generalization of the law of sines to higher dimensions, as defined in F. Eriksson, The law of sines for tetrahedra and n-simplices. However, this generalization misses an important ...

**0**

votes

**1**answer

202 views

### Pasch axiom and Pythagorean field condition?

I am looking for a reference for the claim that the Pasch axiom is equivalent to the Pythagorean field condition, and with respect to what base theory this should be true.
Since posting the question, ...

**10**

votes

**4**answers

763 views

### In which geometries do triangles have an Euler line?

In Euclidean geometry, the centroid, orthocenter and circumcenter of a triangle lie on a line.
In which other geometries does this hold?

**34**

votes

**14**answers

8k views

### Open problems in Euclidean geometry?

Which are some (research level) open problems in Euclidean geometry ?
(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)
I should clarify a bit ...

**1**

vote

**1**answer

64 views

### Set of General Linear Position with Nonzero Measure

I came to the following question, but I don't have quite a good idea how to approach.
Can a set $A\subset \mathbb{R}^n , n\ge 2$ with nonzero measure be in a general linear position?
I believe that,...

**25**

votes

**2**answers

1k views

### Understanding sphere packing in higher dimensions

In a recent publication by the Ukrainian mathematician Maryna Viazovska the Kepler problem for dimension $8$ and $24$, namely the densest packing of spheres, was solved.
Admittedly it is very ...

**4**

votes

**1**answer

265 views

### Sixteen points circle - A conjecture on Möbius plane

The conjecture refer the reader about the Bundle's theorem configuration. (This conjecture from a note)
Consider the Bundle theorem configuration :
Points $A_1, A_2, A_3, A_4$ lie on a circle,
...

**7**

votes

**1**answer

151 views

### Penrose tiling substitution is bijective

Let $\mathcal{P}$ a Penrose tiling built by a substitution $\omega$ with two triangles.
It is claimed, for instance, in the article of Anderson and Putnam "Topological invariants for substitution ...

**1**

vote

**0**answers

165 views

### A chain of six circles associated with six points on a circle (in Mobius plane) [closed]

I found a conjecture: A chain of six circles associated with six points on a circle (in Mobius plane).
This is a generalization of the last my previous question in Three chains of six circles. (...

**3**

votes

**0**answers

172 views

### A conjecture on six planes [closed]

When I read Cox's Theorem, and Clifford's Circle Theorem and Miquel six circles theorem, I found the conjecture as folowing. And I checked the conjecture by the Geogebra sofware, the conjecture is ...

**3**

votes

**0**answers

142 views

### Conjecture generalization of Feuerbach theorem and somes another theorems

My question: I am looking for a solution of a conjecture generalization of the Feuerbach theorem in the end of the topic. But I think, I should let you know why I found this conjecture. I thank to You ...

**6**

votes

**0**answers

125 views

### Paradoxical spherical caps

All spherical caps (i.e. sets $C_L:=\{(x,y,z)|x^2+y^2+z^2=1, z\geq L\}$) admit a paradoxical decomposition in the sense of Banach-Tarski, meaning $C_L \tilde{} 2C_L$; here $\tilde{}$ stands for the ...

**2**

votes

**1**answer

204 views

### Relation of some Euclidean geometry theorems and more conjecture generalizations

In this topic I want to share relation of the Pythagorean theorem, the Stewart theorem and the British Flag theorem, the Apollonius' theorem and the Feuerbach-Luchterhand. Since that I posed two ...

**1**

vote

**1**answer

179 views

### Conjectute: no exist an equilateral triangle such that all vertices are integer numbers [closed]

I am looking for a solution for a conjecture as follows.
In Cartesian plane, no exist an equilateral triangle such that three vertices are integer numbers.
I hope that you like the question ...

**6**

votes

**0**answers

157 views

### Can a closed disc in the plane be partitioned into three disjoint sets which are pair-wise isometric?

Any progress on the following: Can a closed disc in the plane be partitioned into three disjoint sets which are pair-wise isometric, i.e. each set is an image of the others under an isometry?

**3**

votes

**6**answers

1k views

### Circumference of Convex Shapes

Here is a puzzle I found in Mitteilungen der DMV (roughly, "Letters of the German Society of Mathematicians"), issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...

**2**

votes

**1**answer

178 views

### Is it possible to construct an isosceles triangle by using a ruler and without using a pair of compasses?

It is well known that on Euclidean plane one can construct an isosceles triangle on given straight line by using a ruler and a pair of compasses.
Also it is possible to construct straight line ...

**-1**

votes

**1**answer

428 views

### Meeting point of the vertices of a square cloth on x-y plane

Consider a standard square sheet lying on the xy plane with edge length n. Is it possible to determine the coordinates (x, y, z) of the point where the vertices of the sheet will meet, when each of ...

**0**

votes

**0**answers

25 views

### Statistics of Length Eccess in Shortest Path Calculations

I am trying to quantify the error that arises in the following problem:
let $\mathcal{T}$ be a tiling of the plane and the task is to calculate shortest paths in the network $\mathcal{N}$ of the union ...

**4**

votes

**1**answer

128 views

### what's the formula of the inradius of a general simplex? [closed]

As the title, I just want to know whether there is a general formula for calculating the inradius of a n-simplex. Thank you!

**4**

votes

**2**answers

167 views

### Inequality from a point in plane to a triangle OR Inequality on a quadrilateral

If points $A$, $B$, $C$ form a triangle in euclidean space and $D$ is another point in the plane of the triangle, the problem is to show that :
$\frac{AB}{DA + DB} + \frac{BC}{DB + DC} \ge \frac{AC}{...

**7**

votes

**2**answers

311 views

### Closed curve whose neighborhood is as large as possible

Let $C$ be a closed curve in the plane and let $N_\epsilon(C)$ be an $\epsilon$-neighborhood of $C$, like this:
(ignore the fact that the "curve" is polygonal in this picture, I drew it in MATLAB)
...

**2**

votes

**0**answers

110 views

### Which Coxeter groups can be realized as affine reflection groups?

Every affine reflection group has a Coxeter presentation (https://en.wikipedia.org/wiki/Reflection_group#Relation_with_Coxeter_groups). How do you tell which Coxeter presentations arise from affine ...

**16**

votes

**2**answers

829 views

### Probability of two vectors lying in the same orthant

Let $S^{d-1} = \{x \in \mathbb R^d: \|x\| = 1\}$ denote the unit sphere in $\mathbb R^d$. Let $v$, $w$ be drawn uniformly at random from $S^{d-1}$, conditioned on their inner product being equal to $\...

**5**

votes

**0**answers

155 views

### Determining N d-points yielding equal sums of Euclidean distances from M s-points

Given M source points (s-points), determine N, the number of destination points (d-points), and their locations (coordinates), such that the sum of the N Euclidean distances from each source point to ...

**17**

votes

**5**answers

1k views

### Definition of area

I am looking for an attractive, but rigorous definition of area;
say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...

**3**

votes

**1**answer

100 views

### Existence or otherwise of a set of “sufficiently intricate” open sets

Fix $d \in \mathbb{N}$. Do there exist mutually disjoint connected open sets $V_1,\ldots,V_n \subset \mathbb{R}^d$ and $\mathbf{v} \in \mathbb{R}^d$ such that
$\mathbb{R}^d \setminus (\bigcup_{i=1}^...

**5**

votes

**1**answer

408 views

### Is every complex rational algebraic variety simply connected for the Euclidean topology?

Is it true that every quasi-projective rational irreducible algebraic complex variety is simply connected for the Euclidean topology?
Of course, this is false if we replace "complex" with "real" or ...

**0**

votes

**0**answers

65 views

### Maximum value of linear function on the intersection of a parametrized family of balls

Let $C$ be a (nonempty) closed convex subset of $\mathbb{R}^n$ and $a, b \in \mathbb{R}^n$. Using the normal cone characterization of the euclidean projection operator $\mathrm{proj}_C$ (recall that $\...

**3**

votes

**0**answers

63 views

### Equidistribution of Brillouin zones

Answering the question about Limiting shape for Brillouin zones Victor Kleptsyn proved that $N$th Brillouin zone is very close to a circle of radius $c\sqrt N$ (you can find all necessary definitions ...

**12**

votes

**1**answer

363 views

### Limiting shape for Brillouin zones

Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle?
You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...

**54**

votes

**11**answers

7k views

### Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...

**13**

votes

**1**answer

739 views

### A problem in elementary geometry

Let us have a triangle ABC in the Cartesian plane and consider the following transformation of this triangle:
On the ray AB starting at A, select a point B' so that so that |AB'|=|AC|. Likewise,
...

**12**

votes

**3**answers

886 views

### Efficient visibility blockers in Polya's orchard problem

Polya's orchard problem asks for which radius $\rho$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard.
...

**12**

votes

**1**answer

480 views

### Maximal Number of Pairs of Orthogonal vectors in a set of $n$ vectors in $\mathbb{R}^3$

Suppose you are given a set of $n$ non-zero vectors in $\mathbb{R}^3$. What is the maximum number of pairs of them that are orthogonal? The current guess is $\le 2n$.
EDIT: I forgot to add that no ...

**3**

votes

**4**answers

352 views

### Terminology for polygons

As you may know term "polygon" might mean few different things
and its meaning has to guessed from context.
By some reason I have to use few of these meaning in one place.
So I converge to the ...

**7**

votes

**1**answer

966 views

### Geometric meaning of trigonometric relations

According to a paper by Zhiqin Lu in the Mathematical Gazette (the British publication, not the Boston-area newsletter, if that still exists (or even if it doesn't)) in 2007(?), if $u+v+w=\pi$ and $a,...

**0**

votes

**2**answers

106 views

### Pairwise distance distribution for point clouds (normal distribution) [closed]

I have a point cloud (2D for now) of $N$ normally distributed points (with a certain $\sigma$).
My first question would be how the pairwise distance distribution looks (just by chance I discovered a ...

**3**

votes

**1**answer

95 views

### Equality of Euclidean numbers / constructible numbers

Euclidean numbers are those real number that can be constructed from the natural numbers by a finite chain of +,-,*,/ and $\sqrt{}$. They are also called Constructible Numbers.
I am now interested in ...

**8**

votes

**0**answers

175 views

### Ricocheting pinball-like shot: Complexity?

Suppose one has $n$ perfect two-sided mirror segments in the plane $\mathbb{R}^2$.
The segments are open, excluding their endpoints.
They are disjoint as closed segments, i.e., no pair shares an ...

**19**

votes

**6**answers

3k views

### Euclid with Birkhoff

I'm looking for an short and elementary book which does Euclidean geomety with Birkhoff's axioms.
It would be best if it would also include some topics in projective (and/or) hyperbolic geometry.
...