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Questions tagged [euclidean-geometry]

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these.

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238 views

Symmetry group and irreducible representation

Let $S$ be a bounded geometric shape in the Euclidean space $E=\mathbb{R^n}$. Assume that the origin of $E$ is a fixed point of every element of the symmetry group $G(S)$ of $S$, and assume that $G(S) ...
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70 views

Existence of a “generic enough” lattice point interior to a lattice triangle

Let $T$ be a lattice triangle in $\Bbb R^2$ (i.e. the convex hull of three noncolinear points in $\Bbb Z^2$), and assume it has at least one interior lattice point. Is it always possible to find a ...
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159 views

Acute triangles from 100 points

Given $n$ points in general position in the plane, let $P_n$ be the maximum proportion of the $\binom{n}{3}$ triangles with three acute angles. What is the limit $\lim\limits_{n \rightarrow \infty} ...
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72 views

Is the Mandelbrot set weakly self-similar?

A subset $F$ of an Euclidean space $E$ will be called weakly self-similar if for all $x \in F$ there is $\epsilon_x>0$ such that for all positive $\epsilon \le \epsilon_x$ there are $y \in F$, $\...
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1answer
81 views

Algebraic independence of points under isometry

Fix a dimension $n \geq 1$ and a number $k \geq 1$ and suppose that $a_1,...,a_k \in \mathbb{R}^n$ are points in $n$-dimensional space such that among all the $nk$ coordinates there is a subset of ...
2
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141 views

An new equilateral triangle related to the Morley triangle

Morley equilateral triangle is the nice theorem in Eulidean Geometry. I found an equilateral triangle and a group circle related to the Morley triangle and angle trisectors: Let $ABC$ be a triangle ...
3
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206 views

Are these points known? [closed]

Let $ABC$ be a triangle and $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively. From my construction by GeoGebra, I found two special points as ...
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2answers
486 views

Pages from a known textbook on Euclidean geometry?

Do you recall having seen the attached pages in a textbook once? If so, would you be so kind as to share its bibliographic record (or the main items in it) with me below? A teacher provided us xerox ...
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2answers
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Do two new special points in any triangle exist?

There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera. Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $...
29
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2answers
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Why did Dedekind claim that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ hadn't been proved before?

In a letter to Lipschitz (1876) Dedekind doubts that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ had been proved before: quoted from Leo Corry, Modern algebra, German original: Why did Dedekind doubt that $(\...
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24 views

Integer Distances in Pointsets Generated from Simplices with Integer Sidelengths

Let $\mathbb{S}_\mathbb{N}^n$ denote the set of all $n$-simplices in $n$-dimensional euclidean space $E^n$. Call an $n$-simplex aligned, if the set $C$ of its corners satisfies $\exists c_k=\left(x^...
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2answers
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Hyperrectangle that contains most of cube's interior (except its vertices)

Let $n>0$, and let $p,q\in (0,1)$ such that $p<q$. Is there a hyperrectangle $H$ that satisfies the following: $\forall i\in{1,\dots,n}:\\ H\supset \prod_{j=1,\dots,n} \begin{cases} [p,q], &...
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1answer
63 views

Number of Inner Diagonals of Convex Hulls of $n+2$ Points in Convex Configuration in $E^n$

Question: Is it true that $E^2$ is the only Euclidean space, in which the convex hull of $n+2$ points in convex configuration has two inner diagonals and in all other cases there is only one ...
2
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1answer
44 views

Triangle Center from Weighted Perfect Matchings

let $\Delta$ be the triangle whose corners $A$, $B$, $C$ points in general position in Euclidean plane and, let $D$ be a fourth point inside $\Delta$. Question: what is known about the ...
12
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1answer
381 views

Is each cover of the plane by lines minimizable?

A cover $\mathcal C$ of a set $X$ by subsets of $X$ is called $\bullet$ minimal if for every $C\in\mathcal C$ the family $\mathcal C\setminus\{C\}$ is not a cover of $X$; $\bullet$ minimizable if $\...
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43 views

Minimal steps of construction for constructible number

It is known that a real number $\alpha$ is constructible if and only if it lies in a number field $K=K_{n}$ s.t. there exists a tower of field extension $\mathbb{Q}\subset K_{1}\subset K_{2}\subset \...
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112 views

theories where angles exist without a metric

The underlying basic question, which I'm sure I'm not the first to ask, is what are the possible exotic/nonintuitive models of Euclid's axioms/postulates, outside the one where "lines" are interpreted ...
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2answers
240 views

Is it possible to continuously select a probability distribution over fixed points in Brouwer's fixed point theorem?

According to Brouwer's fixed point theorem, for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point. However, these fixed points cannot be chosen ...
5
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What (if anything) is the connection between the Feit-Higman Theorem and the regular plane tilings?

Here are two facts that are superficially similar. Tiling Theorem: The only regular tilings of $\mathbb{R}^2$ are achieved by triangles, squares, and hexagons. Feit-Higman Theorem: The only ...
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1answer
191 views

Is the action of $SO(n)$ on the sphere $S^{n-1}$ ballanced?

A subset $B$ of a group $G$ is called balanced if $gBg^{-1}=B$ for all $g\in G$. An action of a group $G$ on a metric space $X$ is called ballanced if for each non-empty balanced subset $B\subset G$ ...
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1answer
451 views

Is the conjecture true for n-sphere $(n>2)$? [closed]

This is higher dimension conjecture of Problem 3845 in Crux Mathematicorum and Theorem 2 in here: PS: This figure is very nice, this is also generalization of Brianchon’s theorem, The Pascal theorem, ...
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3answers
172 views

Average caliper diameter (mean width) of a polyhedron

Define the caliper diameter of a polyhedron as follows: Let $P_1$ and $P_2$ be two planes both of which are parallel to the x axis such that the perpendicular distance between $P_1$ and $P_2$ is the ...
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1answer
122 views

Controlling angles between vectors using sum of subvector angles?

This is a technical question coming out of my research. Let $\angle(\cdot, \cdot)$ be the angle ($\in [0, \pi]$) between vectors. Consider two vectors $u, v$ in $\mathbb R^3$. Is it true that $$ \...
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25 views

Tangencies of Villarceau circles in a 3D Steiner chain

Consider a Steiner chain made of an arbitrary number $n$ ($\geq 3$) of spheres (not circles, spheres), as in the picture below with $n=6$ (so it is a so-called Soddy hexlet). I've found this picture ...
4
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1answer
367 views

Constructibility of the regular 17-gon [closed]

There is a standard construction of a regular heptadecagon by H.W. Richmond (1893) (https://en.wikipedia.org/wiki/Heptadecagon ) (As anyone knows, it was Gauss who found out that it is possible to do ...
3
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1answer
77 views

Extremum problem on regular simplex

I'm looking for a proof for the extremum problem on regular simplex. Question. Let $\mathcal{A}=A_0A_1...A_n$ be a regular simplex in $\Bbb E^n$. $P$ is a point inside and on boundary of $\mathcal{...
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0answers
49 views

Inequality on simplex with circumscribed sphere

I'm looking for a proof for this problem on simplex which I think it is true Question. Let $\mathcal{A}=A_0A_1...A_n$ be a simplex in $\Bbb E^n$. $(S)$ is circumscribed sphere of $\mathcal{A}$ with ...
2
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2answers
114 views

Centroid and center circumscribed spheres in simplex

I'm looking for a proof for this problem on simplex which I think it is true Question. $A_0A_1...A_n$ is a simplex in the Euclidean space $\Bbb E^n$. $G$ is its centroid and its center ...
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71 views

Coplanar set in metric space

Let $(\Bbb M,d)$ be a metric space. Give three points $X,$ $Y,$ $Z$ in $\Bbb M$ such that they satify one of the following conditions $i)\ d(X,Y)+d(Y,Z)=d(X,Z),$ $ii)\ d(Y,Z)+d(Z,X)=d(Y,X),$ $iii)\...
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171 views

A conjecture on simplex

Let $A_0A_1...A_n$ be a simplex in $\Bbb E^n.$ Let $B_{ij}$ be a point on the edge $A_iA_j,\ 0\le i\not=j\le n.$ Denote by $\beta_i$ the hyperplane passing through the points $B_{i0},$ $B_{i1},$ $B_{...
5
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1answer
163 views

Cyclic quadrilateral in metric space

Consider a metric space $(\Bbb M,d).$ If $X,Y,Z\in \Bbb M.$ We define cosin of angle by $$\cos(\angle YXZ)=\frac{d(X,Y)^2+d(X,Z)^2-d(Y,Z)^2}{2d(X,Y)\cdot d(X,Z)}.$$ If we have four points $A,$ $B,...
4
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1answer
115 views

Pascal's theorem for spherical hexagon

I draw a cyclic spherical hexagon and I check by geogebra that Pascal's theorem is true in this case. My question 1. Is there simple proof for this? My question 2. Can we change the circle on ...
4
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1answer
131 views

Coloring circles in plane

We assume that all the circles in the plane are each colored with one of two colors: red or blue. My question 1. Does there always exist an equilateral triangle such that its circumcircle and its ...
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1answer
79 views

Coloring lines in plane

We assume that all the lines in the plane are each colored with one of two colors: red or blue. Given angle $\alpha.$ My question 1. Is there possible to get two lines with the same color and angle ...
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197 views

A generalization of the Sawayama-Thebault theorem

1. Introduction The Sawayama-Thebault theorem is one of the best nice theorem in plane geometry. The theorem has a long history. It was published in AMM in 1938 the first solution appeared in 1973 ...
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212 views

$N$-$th$ closed chain of six circles

Since 2013, I found a very nice configuration: $N$-th closed chain of six circles. This is a generalization of theorem 1, problem 2 in here and theorem 2 in here and here (and is also generalization ...
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104 views

Pascal theorem for three dimensions

A year ago I found the Pascal theorem for three dimentions as follows: Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1$, $A_2$, $A_3$, $A_4$, $A_5$,...
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1answer
181 views

Yiu's equilateral triangle-triplet points

In more than 2300 years since Euclid's Elements appear, there were only two equilateral triangles become famous: The Morely equilateral triangle and the Napoleon equilateral triangle. In more than ...
6
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1answer
129 views

Example of convex n-gon that cannot be decomposed into k congruent convex polygons

I asked a related question here on MO without any answers yet. The question is in the title - give an example of a convex $n$-gon that cannot be subdivided into $k>1$ congruent convex polygons. ...
4
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1answer
233 views

Reordering vertices of a polygon

Let $Q,Q'$ be two planar polygons with the same number $n>3$ of vertices. There is a correspondence between vertices of $Q$ and $Q'$: to any vertex $z$ of $Q$ corresponds a unique vertex $z'$ of $Q'...
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1answer
300 views

Is there a triangle which makes dense set of angles by drawing medians?

This problem is a restatement of this question, first announced in MathStackExchange. We start with a triangle $T$ in the Euclidean plane and we define $A_n$ as the set of angles of the $6^n$ ...
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16 views

Finding Optimal Spheric Polyhedra with Given Convex Hull Topology

I want to draw finite planar graphs in certain canonical ways. My idea is to use a stereographic projection of the convex hull of points placed on the unit sphere in a way, that the graph induced ...
3
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1answer
81 views

Relative to Isoperimetric inequality with n-polygon

Since Brahmagupta's formula and Bretschneider's formula we have the inequality: Any two quardrilaterals $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ with the same sidelengths and $A_1A_2A_3A_4$ is a cyclic ...
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1answer
377 views

Strengthened version of Isoperimetric inequality with n-polygon

Let $ABCD$ be a convex quadrilateral with the lengths $a, b, c, d$ and the area $S$. The main result in our paper equivalent to: \begin{equation} a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\...
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1answer
52 views

Maximal Vertex Degree of MSTs in Euclidean Spaces

Are there any Euclidean spaces, in which the maximal vertex degree of MSTs (Minimum Spanning Trees) of a finite set of points and edge weights equal to Euclidean distance, isn't equal to the kissing ...
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0answers
29 views

finding subsuming hypervolumes [closed]

Imagine we have an N-dimensional space where each dimension can only have integer values. Imagine further that this space has a set of hypercubes scattered about, each hypercube with its own position ...
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0answers
45 views

Supremum norm of certain quantity II

Can anyone solve the maximization problem...$\max_{|z_i|=1}\Big|\sum_{i,j=1}^nz_iz_j+\sum_{i,j=1}^n|z_i-z_j|\Big|$?
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242 views

A strange planar set and the Continuum Hypothesis

Call a number abnormal if its decimal expansion doesn't feature every digit an infinite number of times. Call a triangle in ${\Bbb R}^2$ abnormal if at least one of its angles spans an abnormal ...
9
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2answers
242 views

Axioms for constructive Euclidean geometry

In the summer I will be teaching a course in (plane) Euclidean geometry to future high school teachers and I am looking for a suitable axiom system (unlike College (Euclidean) geometry textbook ...
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1answer
169 views

Even Isometries in neutral Geometry

Consider a Hilbert plane as in Hartshorne's 'Euclid and beyond' (axiomatic geometry), and its group of isometries f or 'rigid motion' generated by line reflections. Call f 'even' if it is the product ...