The euclidean-geometry tag has no wiki summary.

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### 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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### Is the parallelogram rule an axiom or a theorem in euclidean geometry? [on hold]

I am aware of the proof of the rule in inner product spaces. Excluding the geometry of Descartes, is it possible to prove parallelogram rule or is it an axiom?

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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### 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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### 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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### 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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### 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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### 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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### 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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### Points contained in a disk [closed]

I have a question, but not sure how to prove this.
We are given $n$ points in the Euclidean plane such that there exists no disk of radius $a$ which contains all of the points.
Conjecture: There ...

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### How to find a tetrahedron that covers four points?

I’m looking for an explicit formula for the vertices of a regular tetrahedron that covers four given points. In particular:
Given four distinct real numbers $a_1$, $a_2$, $a_3$, $a_4$, is there a ...

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### 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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### Approximation of a convex body by a contained polytope

This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes.
For a given $\delta$, let $n_\delta$ be the number of faces ...

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### Number of not self-intersecting closed paths spanning $n$ iid uniform points

Let $X_1,X_2,\dots,X_n$ be independent uniform variables in the square. What is the number of piece-wise linear paths which vertices are all the $X_i$ and that do not self-intersect? In other words, ...

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### Successive Inner or Outer Approximation of Simple Polygons with Hierarchies of Implicit Functions

The problem I want to solve, is to quickly decide, whether a point $p=(x^*,y^*)$ is inside or outside of a polygon $P := (p_1, p_2,..., p_n=p_1), p_i := (x_i,y_i)$, with $n$ potentially very large.
...

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### 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, ...

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### Number of small projections

Suppose $X$ is a finite subset of the plane and for $0\leq \theta<\pi$, let $l_\theta$ denote the line through the origin having angle $\theta$ with the positive $x$-axis. For how many values of ...

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### The intersection of $n$ cylinders in $3$-dimensional space

A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such ...

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### Right triangle with edge lengths equal to regular unit polygon edge lengths

This question came up naturally recently from a blog post of John Baez. There is an observation of Euclid that edges of a pentagon, hexagon, and decagon inscribed in a unit circle form the edges of a ...

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### Heronian triangle with two sides that are prime

Can any prime number form a Heronian triangle with a second prime as another side? I cannot find a second prime to form a Heronian triangle with either 23 or 167. I have checked up to the 10^7th prime ...

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### Equipartition of the circle [closed]

Browsing an old technical studies pupil's school book, I have found the description of a method to place at equal distance $N$ points on the circumference of a circle. I am looking for a proof of this ...

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### Is Euclid dead? [closed]

Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" ...

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### 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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### 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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### Shrink polygon to a specific area by offsetting

I have a 2D polygon that I want to shrink by a specific offset (A) to match a certain area ratio (R) of the original polygon. Is there a formula or algorithm for such a problem? I am interested in a ...

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### Is the perimeter of an ellipse with integer axes irrational?

Let $Q$ be an ellipse with integer-length axes $a$ and $b$:
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \;.$$
The perimeter of $Q$ is given by the complete elliptic integral of the 2nd kind, $E(\;)$:
$4 ...

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### Largest inscribed rectangle inside a convex polygon

It has been proved by Radziszewski in this paper
K. Radziszewski. Sur une probleme extremal relatif aux gures inscrites et circonscrites aux gures convexes. Ann. Univ. Mariae Curie-Sklodowska, Sect. ...

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### Random projections and subspace preservation

Consider a set of points $\{\mathbf{x}_i\}_{i=1}^N$ in a $d$-dimensional linear subspace of $\mathbb{R}^n$ (assume $\{\mathbf{x}_i\}$ span the subspace). The points are projected to a $k$-dimensional ...

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### Repeating an operation infinitely makes any convex $n$-gon a regular $n$-gon?

For any convex $n$-gon $P_{0,1}P_{0,2}\cdots P_{0,n}$, let us consider the following operation :
Operation : Let $k=0,1,\cdots$. Take $n$ points $P_{k+1,i}\ (i=1,2,\cdots,n)$ outside of $n$-gon ...

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### About the 'minimum triangle' which includes a convex bounded closed set

Question : Is the following true?
"Letting $K$ be a convex bounded closed set on a plane, then there exists a triangle $M$, which includes $K$, such that $|M|\le 2|K|$. Here, $|M|,|K|$ is the area of ...

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### Finding the probability that $n$ points which are randomly selected from $d-1$ dimensional spherical surface are 'semi-spherical'

Let $S^{d-1}=\{(x_1,\cdots,x_{d})\in {\mathbb R}^{d}|{x_1}^2+\cdots+{x_d}^2=1\}$, and let us call the intersection of any $d-1$ dimensional subset which passes through the origin and $S^{d-1}$ 'a ...

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### Tetrahedron insphere iteration

I know that iterating the following incircle construction approaches an equilateral triangle in the limit:
Starting with any triangle $T$, one forms $T'$ by connecting ...

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### 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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### Soddy-type relation for Steiner chains

For Steiner $n$-chains of circles of radii $r_1,\dots,r_n$ tangent to an inner circle of radius $r_-$ and an outer circle of radius $r_+$, is there a Soddy-type relation between the $n+2$ quantities ...

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### About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon whose area is $S$, then find the max of $\frac{S^\prime}{S}$.
...

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### 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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### About the area of the region where the paper is twofold when you double a piece of paper in the shape of a triangle

Suppose that you have a piece of paper in the shape of a triangle $ABC$ whose area is $S_0$ and that the area of the region where the paper is twofold when you double the paper in two along a line is ...

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### Probability that a stick randomly broken in five places can form a tetrahedron

The following problem was brought to my attention by a doctoral dissertation on Mathematics Education, but - as far as I know - the solution remains unknown.
I have already asked this question on ...

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### Can Tarski decide constructibility in elementary geometry?

Can the decision routine for Tarski's Elementary geometry be extended to decide when an existence claim in that theory can be instantiated by a compass and straightedge construction?
The answer does ...

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### Finding an invisible circle by drawing another line

A friend of mine taught me the following question. He said he found it on a book a few years ago. Though I've tried to solve it, I'm facing difficulty.
Question: You know on a plane there is an ...

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### Conditions for when an off-centre ellipsoid fits inside the unit ball

An ellipsoid $E$ has centre $\vec{c}=(c_1,c_2,c_3)$ and semiaxes $t_1$, $t_2$ and $t_3$ aligned with the $x$, $y$ and $z$ axes. What are the necessary and sufficient conditions on $\vec{c}$, $t_1$, ...

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### 2-layer tilings with a center-of-gravity constraint

I've encountered a tiling problem with a physical constraint that
might place it outside the literature on tiling.
"Tiling" is a bit of a misnomer; it is a special type of cover.
All the tiles are ...

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### Embedding of Two Objects Into Higher Dimensions With Their Sum

Given two vector sets, $\vec x_i$ and $\vec y_i$ (for $i$=1,2,...N, but the dimensionality of each vector can be more than N), let their sum set be $\vec z_i = \vec x_i + \vec y_i$. It's easy to ...