The euclidean-geometry tag has no wiki summary.

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### Did Lucas discover Lucas circles?

MathWord's article on Lucas circles traces the name to a little-known 1973 publication. These interesting circles have found their way into several 21st century publications, including the online ...

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**0**answers

99 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 ...

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votes

**1**answer

66 views

### Geometric realization of an abstract triangulation of the plane

Can every abstract simplicial complex whose geometric realization is homeomorphic to $\mathbb{R}^2$ be realized by a rectilinear triangulation of the Euclidean plane? Alternatively put, can a curvy ...

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**0**answers

94 views

### Examples of Geometric Constructions in Higher Dimensions

The classical problem of geometric construction seems to be restricted to planar Euclidean Geometry with straight edge and compass as the only admissible "construction-tools".
I would like to ...

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**5**answers

448 views

### ten concurrent lines

this relates to a question asked at MSE. i was able to find an answer using complex numbers.
here is the question: there are five points on a circle. take any three points, through the centroid of ...

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**4**answers

612 views

### Isomorphic but non-conjugate subgroups of $GL(n,\mathbb{Z})$ ?

I've been asked some questions by a friend interested in crystallography, and the following questions (I'm not an expert) came spontaneous to me:
1) Are there two finite subgroups ...

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**1**answer

455 views

### What is the correct preposition? (And is there one?)

I just stumbled upon a linguistic problem I wasn't able to resolve via web search. Suppose we're given some geometric set $A$ and subset $B\subset A$. Isn't there a compact way of saying that there ...

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**2**answers

361 views

### Inequality involving the side lengths of a quadrilateral

If $a$, $b$, $c$ and $d$ are the four sides of a quadrilateral, the problem is to show that $ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$. I've verified it to be true for quite a large number of ...

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**3**answers

523 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?

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votes

**1**answer

480 views

### Focus of parabola using only a ruler

It is an easy exercise that using ruler and compass one find the focus of a given parabola.
Can one do the same using only a ruler? -- if not, why?

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vote

**0**answers

92 views

### Equidissection of square [duplicate]

Monsky's Theorem states:
One cannot dissect a square into an uneven number of triangles with equal area.
I was wondering how close one could get to a equidissection, i.e.
For $n$ an uneven ...

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votes

**1**answer

601 views

### The Eyeball Theorem generalized

I have not seen the 2D Eyeball Theorem—that tangents from the centers of two circles, each encompassing the other, intersect each circle in the same segment length—generalized to higher ...

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votes

**1**answer

74 views

### Measuring the Randomness and Statistics of Convex Polygons

How can I tell, how likely it is, that a given convex polygon with a sufficiently high number of edges is random and, if so, what kind of randomness it is (e.g. white noise)?
What is known about ...

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votes

**4**answers

572 views

### Applications of n-dimensional crystallographic groups

I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups)
1) in mathematics
2) outside of mathematics,
besides the applications to ...

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votes

**1**answer

1k views

### Origami Constructions: Intersecting two Circles

It is well known that every construction that can be performed with compass and straightedge alone can also be performed using origami, see:
R. Geretschlager. Euclidean Constructions and the Geometry ...

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**0**answers

266 views

### How can we join two points with a small ruler? [closed]

We want to join by a line two distinct points $A$ and $B$. We have only a ruler of length $l>0$ and a pen. If $AB>l$ how can we do this? Imagine a method that works when $AB$ is really huge and ...

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votes

**1**answer

671 views

### How large can you draw an island on a map?

A cartographer friend asked me this question: could you classify (shapes of) islands by how much space they occupy on a map (comparatively to how much space is occupied by water) if you draw them as ...

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votes

**1**answer

62 views

### Calculating the “Belvedere Hull” of a Simple Planar Polygon

As an informal motivation the problem, imagine a tower with polygonal footprint, that is located in a beautiful landscape, the "Belvedere Hull" is then related to the directions, in which one would ...

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**2**answers

656 views

### Term for “uncheckable constructions”

Is there a term for "uncheckable geometric constructions"?
Say, Angle Trisection and Doubling the Cube are checkable;
i.e., if the answer is given one can do finite Compass-and-straightedge ...

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votes

**1**answer

217 views

### Dubins car shortest paths: Decidable?

A Dubins car follows a
Dubins path
in $\mathbb{R}^2$, with constant wheel speed and
limited turning radius.
It is known that the shortest Dubins path in the absence
of obstacles follows circular
arcs ...

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votes

**1**answer

125 views

### Three-dimensional Apollonian spirals

Given mutually (externally) tangent spheres $S_1$, $S_2$, $S_3$, $S_4$, let $S_n$ be the unique sphere externally tangent to $S_{n-1}$, $S_{n-2}$, $S_{n-3}$, and $S_{n-4}$ for $n \geq 5$.
Let ...

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**4**answers

190 views

### Inside-out polygonal dissections

A dissection of a polygon $P$
is a partition of $P$ into a finite number of pieces, which can then be rearranged
(via planar translations and rotations) and joined (without overlap) to form a new ...

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votes

**1**answer

196 views

### Generalized Hlawka inequality

Let $E$ be a vector space over the real (the complex case is interesting too). We consider functions $f:E\rightarrow\mathbb R$ which satisfy the homogeneity property
$$f(\lambda x)=|\lambda|\,f(x).$$
...

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votes

**2**answers

320 views

### Are angles between points enough to decide the realizability?

Let n points in the plane be given whose coordinates we don't know.
Assume, however, that for any triple of the points we know the angle.
Question: Can we decide whether the n points are realizable ...

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**4**answers

357 views

### What can be said of the structure of a metric space without isosceles triangles?

This is a question that has been bothering me in the back of my head for quite some time.
Suppose we have a metric space $X$ with metric $\mathrm{d}$. By an isosceles triangle we mean a tuple of ...

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**1**answer

56 views

### Existence of half-planes with respect to regular open sets of the Euclidean plane

I initially asked this question at math.stackexchange.com but there was no reaction, so I thought this may be a good idea to transfer it to mathoverflow.net
Let ...

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**6**answers

1k views

### Tetrahedra with prescribed face angles

I am looking for an analogue for the following 2 dimensional fact:
Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is ...

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votes

**1**answer

1k views

### Finding a minimum bounding sphere for a frustum

I have a frustum (truncated pyramid defined by six planes) and I need to compute a bounding sphere for this frustum that's as small as possible.
I can choose the centre of the sphere to be right in ...

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votes

**1**answer

575 views

### Malfatti Circles - Limiting point

"Three circles packed inside a triangle such that each is tangent to the other two and to two sides of the triangle are known as Malfatti circles" (for a brief historical account on this topic, see ...

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votes

**4**answers

813 views

### How to compute the average distance till intersection within a triangle in R^2?

Lots of simple questions because I am a noob.
You are given 3 points in R^2; A, B, C forming a triangle with area > 0. You pick an arbitrary point inside ABC and an arbitrary direction. After some ...

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**1**answer

165 views

### Random non-intersecting circles in the plane

If I give a finite region of $\mathbb{R}^{2}$ and place $k$ circles of radius $r(k)$ uniformly at random inside, are there any known results for the probability that the circles do not overlap? ...

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**0**answers

76 views

### Containing a “fuzzy” ellipsoid within an ordinary ellipsoid

Consider the ellipsoid described by the inequality $(x - x_c)^T P^{-1} (x - x_c) \leq 1$, where the vector $x_c \in \mathbb{R}^n$ denotes the center of the ellipsoid and the symmetric positive ...

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votes

**1**answer

935 views

### How to find the Fermat Point using the construction of the tangent to ellipse?

Be done the triangle ABC, it is known the method to finding the point Q that minimises the sum QA+QB+QC among all points Q in the plane (The Fermat point).
I want a hint for solving this problem using ...

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### Side-Angle-Side Congruence and the Parallel Postulate

Is there a link between the side-angle-side congruence of triangles and the parallel postulate? Specifically, does it follow from Euclid's first four axioms alone? In fact, does it even follow from ...

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**1**answer

211 views

### Maximum number of general-position points with mutual rational distances?

Richard Guy has shown that there are six points in the plane—no three collinear,
no four cocircular—such that all interpoint distances are rational.
Guy, Richard. Unsolved Problems in ...

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votes

**2**answers

424 views

### Historical question re: ellipses obtained by certain geometrical constructions

I am a faculty member in the Forensic Science Program at PennState (UP). I am trying to obtain information of a historical nature concerning two closely related topics. I seek historical references ...

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**2**answers

112 views

### Maximum possible number of similar three-colored triangles

I want to maximize the number of similar triangles with vertices from three fixed sets, one vertex from each set. For example, if you fix two points $X$, $Y$ (i.e. two sets with only one member), then ...

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votes

**4**answers

865 views

### Metrics for lines in $\mathbb{R}^3$?

I seek a metric $d(\cdot,\cdot)$ between pairs of (infinite) lines in $\mathbb{R}^3$.
Let $s$ be the minimum distance between a pair of lines $L_1$ and $L_2$.
Ideally, I would like these properties:
...

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**9**answers

6k 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 ...

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**2**answers

133 views

### Hexagon Formed by connecting Trisections of triangle sides [closed]

Is there a theorem for the area of the hexagon formed by connecting the points formed when the sides of a triangle are trisected? It appears that the ratio of the area of the triangle to the area of ...

**2**

votes

**1**answer

226 views

### Tools for Removing Radicals from Equations

I am currently doing some investigations on Sylvester's 4 Point Problem Probability of 4 Points being in Convex Configuration
and repeatedly face the problem of solving equations between sums of ...

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votes

**3**answers

696 views

### On maximal regular polyhedra inscribed in a regular polyhedron

Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1.
For example, it's easy to prove that ...

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**0**answers

99 views

### A conjecture about cross sections of a pyramid [closed]

Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon.
This is a conjecture I came across while trying to solve this problem.
I was ...

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**3**answers

1k views

### Angle of a regular simplex

I find the following question embarrassing, but I have not been able to either resolve it, or to find a reference.
What is the vertex angle of a regular $n$-simplex?
Background: For a vertex $v$ ...

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**0**answers

161 views

### Continuity of minimizers to distance function from point to convex set

Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$):
$\min_{x\in U}\|x-y\|$.
I believe the minimizer $x_{U}^{*}$ is ...

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votes

**1**answer

255 views

### Conditions for a parametric curve to avoid self-intersection?

Suppose a planar curve $C$ is defined by parametric
equations in $t$: $x(t)$ and $y(t)$.
Are there conditions on these two functions that guarantee
that $C$ does not self-intersect?
For example,
the ...

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votes

**1**answer

272 views

### Generalising Euler's formula to ellipses and three dimensions

Let $D$ be the closed unit disk, $T$ a triangle and $E$ an ellipse with $E\subset T \subset D$. Without loss of generality say that $E$ is centred at Cartesian coordinates $(c, 0)$ with $0\leq c \leq ...

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**0**answers

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### Probability of 4 Points being in Convex Configuration

Background of my question is, that I would like to implement a parallel preprocessing for a constructing the convex hull of very huge number of points in the euclidean plane;
the idea is to process ...

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**1**answer

426 views

### Reference request: a differential equation in elementary geometry

15 hours and four up-votes but not a word from anybody. That's the result of this question to stackexchange.
My question is where the following differential equation arises naturally and where it ...

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votes

**1**answer

3k views

### spiral of Theodorus

A long time ago when I was in college I read about making a spiral out of right triangles with sides 1 and $\sqrt{N}$. (A google search seems to indicate that this is called the Spiral of Theodorus.)
...