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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6 votes
1 answer
215 views

Necessary and sufficient condition for tangential polygon to be cyclic

Can you prove or disprove the following claim? Claim. Let $A_1,A_2, \ldots ,A_n$ be the vertices of an $n$-sided tangential polygon and let $B_1,B_2, \ldots ,B_n$ be the contact points of the ...
5 votes
0 answers
481 views

Looking for journal (without fees) to publish a research paper in Euclidean geometry

I am looking for a place to publish a research paper in Euclidean geometry. This is a fairly lengthy article (56 pages) in which I present a fundamental property of polygons. I have already been ...
5 votes
1 answer
299 views

Embedding an icosahedron

A transitive set in $\mathbf{R}^n$ is a finite set with a transitive group of symmetries. I want to understand how subsets of a transitive set constrain the group. Let me start with the example of a ...
4 votes
1 answer
211 views

Point of concurrency [closed]

I am looking for the proof of the following claim: Claim: Let $\triangle ABC$ be an arbitrary triangle, $D$ its nine-point center and $E,F,G$ are the nine-point centers of the triangles $\triangle ...
5 votes
1 answer
234 views

Smallest regular $m$-gon covering a regular $n$-gon

I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question. Let us fix a regular $n$-gon with area $1$. What is the smallest ...
4 votes
3 answers
280 views

Finding Pythagorean quadruples on a given plane?

In 2D one cannot construct Pythagorean triples $x^2+y^2=m^2$ ($x,y,m\in\mathbb{Z}$) that lie on every line through the origin (e.g., a Pythagorean triple with $x=y$ would require $\sqrt{2}$ to be ...
9 votes
0 answers
328 views

Which tetrahedra are scissor congruent to a cube?

Question: Which Euclidean tetrahedra are scissor congruent to cubes? Consider a Euclidean tetrahedron $T$ in $\mathbb{R}^3$ with edge lengths $l_1,\ldots, l_6$ and dihedral angles $\alpha_1,\ldots, \...
16 votes
1 answer
981 views

Status of Larry Guth's Sponge Problem

[Edited Jan 23, 2021] Let $D^n$ be the $n$-dimensional unit radius disk in euclidean $\mathbb{R}^n$. Larry Guth's Sponge Problem asks: Does there exist a constant $\epsilon=\epsilon_n$ such that every ...
2 votes
0 answers
111 views

Sufficient coordinate-free condition for points being co-spheric

Question: is there a theorem that guarantees that $\mathcal{P}\subset\mathbb{E}^n$ is finite set of points in a Euclidean space and all radii of the $(n-1)$-spheres that are defined by the $n$-...
1 vote
1 answer
297 views

A generalization of Harcourt's theorem

This question is closely related to my previous question. Can you prove the claim given below? The following claim is a conjectured generalization of Harcourt's theorem. Claim. Let $A_1,A_2 \ldots ...
5 votes
4 answers
554 views

Optimizing the gradient norm on the unit sphere

Let $ \Bbb S^{d-1}=\{(x_1,\cdots ,x_d): x_1^2+ \cdots +x_d^2=1\}\subset \Bbb R^d$ be the unit sphere. Let $\nabla u= (\partial_{x_1}u,\cdots, \partial_{x_d}u)$ be the gradient of a function $u\in C_c^\...
6 votes
0 answers
311 views

Does this plane geometry theorem have a name (well-known)?

Consider three circles $(O_1)$, $(O_2)$, $(O_3)$. Denote the homothetic center of $\{$$(O_1)$, $(O_2)$$\}$ by $A$, the homothetic center of $\{$$(O_2)$, $(O_3)$$\}$ by $B$. Let $C$, $D$ be two points ...
1 vote
1 answer
292 views

A formula for the area of bicentric quadrilateral

Can you provide a proof for the claim given below? The following claim is inspired by Harcourt's theorem and can be seen as its generalization to quadrilaterals. Claim. Given bicentric quadrilateral $...
5 votes
0 answers
242 views

Are there any neusis-hard/neusis-complete problems?

I have lately been enjoying Richeson's Tales of Impossibility (see MAA review), an accessible book on the famous problems of Euclidean geometry including angle trisection/cube doubling/heptagon ...
80 votes
1 answer
3k views

Converse to Euclid's fifth postulate

There is a fascinating open problem in Riemannian Geometry which I would like to advertise here because I do not think that it is as well-known as it deserves to be. Euclid's famous fifth postulate, ...
2 votes
1 answer
192 views

The centroid, the first and second Napoleon points and $X(930)$ lie on a circle

Can you provide an elementary proof for the claim given below? Preliminary definitions: $X(110)=$ focus of Kiepert parabola. $X(137)=X(110)$ of orthic triangle . $X(930)=$ anticomplement of $X(137)$ . ...
1 vote
0 answers
88 views

Differential of the gradient of a strictly convex function

For $n\geq 2$, we consider $\mathbb{R}^n$ endowed with the usual scalar product. Let $f\in\mathcal{C}^2(\mathbb{R}^n,\mathbb{R})$ be a striclty convex function such that $\nabla f$ is nowhere ...
2 votes
1 answer
293 views

Expected triangle area of normal distributed vertices with colinear expectations

For the bounty the already answered problem was reformulated This question was already answered for random variables in $\mathbb{R}^3$. Now I am looking for the solution in $\mathbb{R}^2$ that could ...
6 votes
0 answers
222 views

What is the expected value of the volume of a tetrahedron inscribed in the unit sphere?

Four (non-coincident) points on the unit sphere determine a tetrahedron. What is the expected value of the volume of such a tetrahedron--the volume of the sphere itself being $\frac{4 \pi}{3} \approx ...
1 vote
0 answers
49 views

How to tile a plane such that moving from one tile to the next in any of the 8 cardinal directions is the same length?

When tiling the euclidean plane with squares (like most board games), moving diagonally to another tile is longer than moving vertically or horizontally. Is there a tiling such that moving in any of ...
2 votes
1 answer
252 views

Checking planar convexity of 4 points with Stewart's formula

Is the following conjecture correct? Conjecture: If $A,B,C,D$ are four points in general position in the euclidean plane, with $a:=\|C-B\|,\ \ b:=\|C-A\|,\ \ c:=\|B-A\|$ $a':=\|D-A\|,\...
3 votes
1 answer
275 views

How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle

How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle? See also: Malfatti circles
3 votes
1 answer
769 views

Brother of Japanese theorem for cyclic quadrilaterals

I am looking for a proof of a like result as follows and Higher-dimensional generalizations? Let $A, B, C, D$ be four point with lengths of $AB, BC, CD, DA$ are $a, b, c, d$ respectively. Let $F \in ...
0 votes
0 answers
92 views

Lines through the origin every pair of which meet at the same angle

This item isn't getting attention, so I'll try it here: begin quote The three lines through antipodal pairs of centers of faces of a cube meet each other pairwise at $90^\circ$ angles. The three lines ...
6 votes
1 answer
406 views

Algorithm to decide whether two constructible numbers are equal?

The set of constructible numbers https://en.wikipedia.org/wiki/Constructible_number is the smallest field extension of $\mathbb{Q}$ that is closed under square root and complex conjugation. I am ...
1 vote
0 answers
34 views

Maximizing the volume of the intersection of a fixed ball with a cube with varying width and location

Given a ball $B$ and a linear subspace $L$ in $\mathbb{R}^n$, what is the maximum value of $\frac{vol(B \cap C)}{vol(C)}$ where $C$ is a cube of the form $x + [0, h]^n$ for $x \in L$ and $h \in \...
54 votes
5 answers
2k views

Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra

Suppose you have a tetrahedron $T$ in Euclidean space with edge lengths $\ell_{01}$, $\ell_{02}$, $\ell_{03}$, $\ell_{12}$, $\ell_{13}$, and $\ell_{23}$. Now consider the tetrahedron $T'$ with edge ...
2 votes
0 answers
79 views

Can an exterrior of a ball in Euclidean space be considered a ball itself under any proposed generalization?

If we take an n-dimensional Euclidean space and cut off a ball centered at origin, we get a set that has boundary equal to the surface area of the cut off ball. I wonder whether there were any ...
8 votes
0 answers
204 views

Which subsets of the plane are similar to all their affine images?

A parabola P in the plane has the nice property that the image of P under any affine transformation is similar to P itself. Which other subsets of the plane have this property? I wondered aloud about ...
3 votes
1 answer
133 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 ...
35 votes
6 answers
2k views

Trigonometry / Euclidean Geometry for natural numbers?

Let $d(a,b) = 1 - \frac{2\gcd(a,b)^3}{ab(a+b)}$ be a metric on natural numbers without $0$. The metric space $X = \{x_0,x_1,\cdots,x_n\},n>2$ is isometric embeddable in $\mathbb{R}^n$ if and only ...
3 votes
1 answer
103 views

Solid angles at points in an orthosimplex

Given a point ${\bf x} = (x_1,x_2,\dots,x_n)$ in the orthosimplex $K = \{(x_1,x_2,\dots,x_n)\ : \ 0 \leq x_1 \leq x_2 \leq \dots \leq x_n \leq 1\}$, what proportion of a ball of radius $\epsilon$ ...
5 votes
0 answers
299 views

Trade-off between covering number, ball radius and diameter of $d$-dimensional shapes

Given any $d$-dimensional shape $X$ in the Euclidean space, let $\ell(X)$ be the length of the longest line segment connecting two points of $X$. How can we prove the following statement? There exists ...
4 votes
1 answer
358 views

Trade-off between hypervolume and diameter of $d$-dimensional shapes having a hypercubic smallest bounding box

Given any $d$-dimensional shape $X$, let $V(X)$ be its $d$-dimensional volume, and let $\ell(X)$ be the length of the longest line segment connecting two points of $X$. Let $\mathcal{S}_C$ be the set ...
7 votes
1 answer
152 views

Can prolates overlap more easily than oblates?

Context: When modeling anisotropic particles, the two common types of shapes of interest are cylindrical and disk-like particles. For simplicity let us say we model these as prolates and oblates ...
1 vote
0 answers
40 views

If the volume-ratio of an inscribed convex set to the circumscribing convex set is rational, can anything of consequence be further deduced?

Say, one has two $n$-dimensional convex sets $A$ and $B$, with $B$ being inscribed in the strictly larger set $A$. ($A$ and $B$ have at least one boundary point in common. $B$ “fits snugly” in $A$ ...
3 votes
0 answers
69 views

On isospectral planar domains (and a paper by Buser, Conway, Doyle and Semmler)

I have never seen a short, elegant way (from the viewpoint of a non-topologist) which constructs isospectral planar domains from Sunada group triples, although essentially those triples live at the ...
2 votes
1 answer
96 views

There is no general method to construct n-regular polygon such that the given n-polygon inscribed the n-regular polygon

Conjecture 1: With $n\ge 5$, given general n-polygon, there is no general method to construct n-regular polygon such that the given n-polygon inscribed the n-regular polygon (with one and only one ...
3 votes
0 answers
57 views

A canonical map from a Euclidean cone-manifold $M^3$ to $\mathbb{E}^3/\mathrm{Hol}(M)$

Suppose we have a 3-dimensional Euclidean cone-manifold $M$—in my book that just means $M$ is a manifold whose geometry is constructed by gluing it out of Euclidean tetrahedra, with faces paired by ...
12 votes
1 answer
1k 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 ...
1 vote
0 answers
82 views

How can construct the equilateral $A''B''C''$ such that area of $A''B''C''$ is biggest

Let $ABC$ be arbitrary triangle in a plane. Let $A'B'C'$ and $A''B''C''$ be two equilateral triangles such that $A \in B'C'$, $B \in C'A'$, $C \in A'B'$ and $A \in B''C''$, $B \in C''A''$, $C \in A''B'...
3 votes
1 answer
487 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 ...
1 vote
2 answers
153 views

ratio between a polygon bounded in another polygon [closed]

Let A be a convex polygon with area SA. Construct a new polygon B by orderly connecting the midpoints of the segments of A. Denote the area of B by SB. Claim : the ratio SB/SA is constant for all ...
5 votes
2 answers
344 views

Frégier and Frégier's Theorem

A curious and interesting gem is Frégier's theorem, quoted here from David Wells: Choose any point $P$ on a conic, and make it the vertex of a right angle which rotates about $P$. Then the ...
18 votes
2 answers
1k views

Emergence of the orthogonal group

Do we know what mathematician first considered, and perhaps named, what we call the group $\mathrm O(n)$, or $\mathrm{SO}(n)$, for some $n>3$? I mean it specifically as group (not Lie algebra) ...
4 votes
0 answers
218 views

Studying finite groups with Euclidean geometry?

Since each finite group $G$ can be considered as a subgroup of the symmetric group, by Cayley's theorem, we might see the elements of $G$ as permutations $\pi$. Consider for each $\pi \in G$ the set: ...
4 votes
1 answer
208 views

Is this elementary formula for the parabolic segment new?

Recently (May 2020) a formula for the area of the parabolic segment (i.e. the region enclosed by a parabola and a line), in terms of the coefficients of the Cartesian equations, has been published by ...
60 votes
1 answer
6k views

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

Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a foot-...
0 votes
0 answers
103 views

Comparing Euclidean norm of two normal vectors

Let $X_i$ ($i = 1,2$) be two random vectors in $\mathbb R^n$, with normal distribution with scalar covariance matrix $\sigma_i^2$ and center $\mu_i$ (in my case, $n = 2$). Is there a way to estimate ...
27 votes
1 answer
979 views

The lion and the zebras

The lion plays a deadly game against a group of $N$ zebras that takes place in the steppe (= an infinite plane). The lion starts in the origin with coordinates $(0,0)$, while the $N$ zebras may ...

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