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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Equal products of triangle areas

Can you prove the following claim: Claim. Given hexagon circumscribed about an ellipse. Let $A_1,A_2,A_3,A_4,A_5,A_6$ be the vertices of the hexagon and let $B$ be the intersection point of its ...
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1answer
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Cramer–Castillon problem like

Cramer–Castillon problem being very difficult problem. Related to the Cramer–Castillon problem configuration, I posed a problem as follows: Let $ABC$ be arbitrary triangle and let three collinear ...
1answer
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Equal sums of line segments

I would like to see a proof of the following Claim. Let $A_1,A_2,A_3,A_4,A_5$ be vertices of bicentric pentagon. Let $B_1$ be the intersection point of $A_1A_3$ and $A_2A_5$, $B_2$ the intersection ...
0answers
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Analytic lower-bound for minimal value of $\|x\|^2$ such that $\|Cx-b\|^2 \le c^2$ (a hyperellipsoid)

Let $C$ be an $n \times p$ matrix and $b$ be a column vector of length $n$, and $c>0$. Let $E := \{x \in \mathbb R^p \mid \|Cx-b\| \le c\}$, a hyperellipsoid in nonstandard position. Question 1. ...
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Covering the sphere with an approximately planar grid

Consider a triangulation of a radius $R$ sphere into $n$ triangles. Must $Ω(\sqrt n)$ triangles have $Ω(1)$ relative difference from being an equilateral triangle of area $4πR^2/n$?  ($Ω$ is from ...
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Generalization of the half-angle formulas

The following is a generalization of the half-angle formulas presented in the following link for a triangle: http://www.nabla.hr/GE-AppTrigonomB1.htm Generalization. Let $a$, $b$, $c$, $d$ be the ...
2answers
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Which knots appear as the singular locus of a polyhedral metric on the 3-sphere?

What can be said about a knot $K\subseteq S^3$ for which there exists a (Euclidean) polyhedral metric (aka Euclidean cone-manifold structure) on $S^3$ whose singular locus is precisely $K$? I'm ...
2answers
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Necessary and sufficient condition for quadrilateral to be cyclic

Can you provide a proof for the following proposition: Proposition. Given any quadrilateral $ABCD$. Let $P,Q,R,S$ be nine-point centers of triangles $\triangle ABD$,$\triangle ABC$,$\triangle BCD$ ...
2answers
710 views

Three circles intersecting at one point

Can you provide a proof for the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with nine-point center $N$ and circumcenter $O$. Let $A',B',C'$ be a reflection points ...
1answer
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Collinearity of three significant points of bicentric pentagon

Can you provide a proof for the following claim? Claim. Given bicentric pentagon. Consider the triangle whose sides are two diagonals drawn from the same vertex and side of pentagon opposite from ...
1answer
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Most natural definition of Euclidean geometry [closed]

What is the "least" amount of structure in terms of axioms and assumptions that is needed to define a Euclidean geometry. For example, is any set {p} a with a not necessarily explicitly ...
2answers
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Can the fugitive escape?

A fugitive is surrounded by $N$ police officers, with the nearest one at distance $1$ away. The fugitive and the officers move alternatively. In a fugitive move, the fugitive can travel no more than ...
1answer
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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 ...
1answer
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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 $K$-...
1answer
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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 ...
0answers
142 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 ...
1answer
137 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 ...
1answer
211 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 ...
2answers
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1answer
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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 $X(137)$ . ...
0answers
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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 ...
1answer
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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 ...
1answer
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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 ...
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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 ...
3answers
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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 ...
1answer
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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 ...
1answer
673 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 $AC$ , ...
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What are the expected values of the volumes of two classes of ellipsoids contained within the unit 3-ball, and/or what is their ratio?

Consider the class of all ellipsoids contained in the unit 3-ball, and also the subclass of those ellipsoids also contained within tetrahedra also contained in the unit 3-ball. What are the expected ...
1answer
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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 ...
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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 ...
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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 ...
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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 ...
1answer
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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 $\epsilon$ ...
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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 ...