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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Is Tarskian hyperbolic geometry consistent, complete & decidable?

Tarski developed an axiomatic description of Euclidean geometry in first order logic. Its primitive notions are points and its primitive relations are betweeness and congruence of points. The Parallel ...
Mozibur Ullah's user avatar
11 votes
1 answer
212 views

The set of boundary vectors of compact convex body has empty interior

Let $K$ be a compact convex body in the Euclidean space $\mathbb R^n$ and $\partial K$ be its topological boundary in $\mathbb R^n$. Definition. A vector $\mathbf v\in\mathbb R^n$ is called $K$-...
Taras Banakh's user avatar
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4 votes
1 answer
286 views

Collinearity in bicentric polygons

Can you provide a proofs for the following two claims? Claim 1. The circumcenter, the incenter, and the intersection of the principal diagonals in a bicentric even-sided polygon are collinear. Claim ...
Pedja's user avatar
  • 2,673
5 votes
0 answers
480 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 ...
Jesús Álvarez Lobo's user avatar
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 ...
Pedja's user avatar
  • 2,673
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 ...
Sean Eberhard's user avatar
1 vote
2 answers
307 views

Is there an area-preserving concentric diffeomorphism of the ellipse?

$\DeclareMathOperator\Vol{Vol}$This is a cross-post. Let $0<b<1$ be a fixed parameter, and let $(R(\theta),\theta)$ be the polar coordinates of the ellipse $$E=\{(x,y) \in \mathbb R^2 \, | \, \...
Asaf Shachar's user avatar
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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 ...
Pedja's user avatar
  • 2,673
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 ...
Luis Ferroni's user avatar
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2 votes
0 answers
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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 $n$-...
Manfred Weis's user avatar
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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 ...
Pedja's user avatar
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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 ...
Đào Thanh Oai's user avatar
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 $...
Pedja's user avatar
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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 $X(137)$ . ...
Pedja's user avatar
  • 2,673
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 ...
G. Panel's user avatar
  • 557
2 votes
1 answer
173 views

Four concyclic triangle centers

Can you prove the claim given below? Inspired by Lester's theorem I have formulated the following claim: Claim. Given any scalene triangle $\triangle ABC$ . Let $D$ be the reflection of incenter in ...
Pedja's user avatar
  • 2,673
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 ...
JHM's user avatar
  • 2,254
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 ...
Mark S's user avatar
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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 ...
Jim McCann's user avatar
2 votes
2 answers
520 views

A generalization of Napoleon's theorem

Can you provide a proof for the following proposition? Proposition. Given an arbitrary $\triangle ABC$. The $\triangle AEB$, $\triangle BFC$ and $\triangle CDA$ are constructed on the sides of the $...
Pedja's user avatar
  • 2,673
2 votes
2 answers
243 views

Six concyclic points

Can you provide a proof for the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with excenters $J_A$,$J_B$ and $J_C$ . Let $G$ be the orthogonal projection of the $...
Pedja's user avatar
  • 2,673
3 votes
2 answers
243 views

Four concyclic points inside bicentric quadrilateral

Can you provide a proof for the following proposition: Proposition. Let quadrilateral $ABCD$ be inscribed into a circle with center $O$ and circumscribed around a circle with center $I$. Let $X$ be a ...
Pedja's user avatar
  • 2,673
12 votes
2 answers
915 views

Intersection point of three circles

Can you provide a proof for the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with orthocenter $H$. Let $D,E,F$ be a midpoints of the $AB$,$BC$ and $AC$ , ...
Pedja's user avatar
  • 2,673
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 ...
granular_bastard's user avatar
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 ...
Paul B. Slater's user avatar
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 ...
YEp d's user avatar
  • 11
2 votes
2 answers
257 views

Minimum Euclidean squared norm in the convex hull of points with rational coordinates

This is probably known, but I have not located a reference. Let $P$ be the convex hull of $k$ points in $\mathbb R^n$ with rational coordinates. Consider the Euclidean square norm function $F:P\to\...
Claudio Gorodski's user avatar
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 ...
Michael Hardy's user avatar
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 \...
pinaki's user avatar
  • 5,064
90 votes
5 answers
4k views

Does this property characterize straight lines in the plane?

Take a plane curve $\gamma$ and a disk of fixed radius whose center moves along $\gamma$. Suppose that $\gamma$ always cuts the disk in two simply connected regions of equal area. Is it true that $\...
Alessandro Della Corte's user avatar
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 ...
Anixx's user avatar
  • 9,306
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 ...
Robin Houston's user avatar
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 $\epsilon$ ...
James Propp's user avatar
  • 19.4k
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 ...
Penelope Benenati's user avatar
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 ...
user929304's user avatar
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 $A$ ...
Paul B. Slater's user avatar
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 ...
Penelope Benenati's user avatar
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 ...
Đào Thanh Oai's user avatar
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 ...
THC's user avatar
  • 4,313
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^\...
Guy Fsone's user avatar
  • 1,033
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 ...
Đào Thanh Oai's user avatar
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 ...
Tom Sharpe's user avatar
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'...
Đào Thanh Oai's user avatar
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 ...
Arie Rokach's user avatar
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
Đào Thanh Oai's user avatar
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) ...
Francois Ziegler's user avatar
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: ...
user avatar
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 ...
archimedes_segment's user avatar
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 ...
Circonflexe's user avatar
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 ...
Eric's user avatar
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