# 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 Morley’s observation complete?

Morley’s observation states that in a triangle the intersections of trisectors proximal to a (triangle) side lie six by six on three triples of parallel lines that make angles of 60° with each other. ...
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### What sequence maximizes the final distance?

This problem was created by professor Ronaldo Garcia from Universidade Federal de Goiás (UFG) and he showed it to me at an event in my university. This problem has a lot of history and he told me he ...
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### $1$-Lipschitz map from hyperbolic to Euclidean plane

I'm trying to find a reference to the following statement. Define a function $f$ from the hyperbolic plane (in the Poincaré unit disc model using polar coordinates) to the Euclidean plane (using polar ...
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### Alternative equivalence results for the constructibility of real numbers

Everyone is aware of the standard result from undergraduate field theory that a real number $\alpha$ is constructible by straightedge and compass if and only if there exists a finite sequence of field ...
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### Minimum diameter of set inscribed in a unit sphere

For a study of the stability of certain maps taking values in a sphere I have the following question. Let $A$ be a subset of $\mathbb{R}^n$. Suppose $A$ lies in a unit ball, but in no ball of smaller ...
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### Condition to be concyclic [closed]

What condition would you impose upon $n$ points on a plane of which no three points are collinear so that they are concyclic if the distances of each point from the all remaining points are known? (...
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### How many distances are required to calculate all distances among $n$ points in the Euclidean plane?

I want to know all the pairwise distances between points $P_1,P_2,\ldots,P_n$ in the Euclidean plane (or equivalently, I want to reconstruct the set of points up to congruence). Let's say I have an ...
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### Symmetrically $3d$ embeddable graphs with high girth

Let a symmetrically $3d$ embeddable graph be a graph that can be embedded into $3d$ so that the embedding is arc-transitive, which means that every vertex-edge pair with the vertex incident to the ...
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### Is "Escherian metamorphosis" always possible?

$\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$This is a tweaked ...
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### A textbook on foundations of geometry in spirit of Tarski

I am interested in a textbook for studying (and teaching) foundations of geometry in the spirit of Tarski. I know that there is a rather old German book [W. Schwabhäuser, W. Szmielew, A. Tarski, ...
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### Orthogonal projection of a point centrally-symmetric closed convex subset of $\mathbb R^n$ never expands the coordinates of the point

Let $C$ be a closed convex subset of $\mathbb R^n$ which is symmetric about the standard coordinate axes. For example, think of $C$ as the unit-ball for an $\ell_p$-norm, for some $p \in [1,\infty]$. ...
1 vote
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### A 'natural' enumerable metric space with integral distances which is essentially the Euclidean space

It is easy to construct a metric space $E_d$ such that all points of $E_d$ are at mutually integral distance and such that there is a map $\varphi$ from $E_d$ into the $d$-dimensional Euclidean space ...
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### How many equilaterals have vertices intersections of angle trisectors of a triangle?

The celebrated Morley’s theorem ensures that the interior trisectors, proximal to sides respectively, meet at vertices of an equilateral. In the paper Trisectors like Bisectors with Equilaterals ...
1 vote
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### Fashioning higher precision tool from a lower precision tool

I'm not sure where to ask this question. Suppose I have one or more rulers with which I can measure distances with up to 1 mm error. Is there a way I could make another tool of higher precision (e.g. ...
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### Generalization of IMO5 from 1987

The following question appeared as question 5 on the IMO in 1987: Prove that for all $n \geq 3$ one can find $n$ distinct points on the Euclidean plane with the property that the distance between any ...
1 vote
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### A metric geometry problem which calculates the limitation of human eyes

This is the update version of this question A functional inequality which calculates the limitation of human eyes Let an Euclidean space $M$ (or a path connected metric space) be partitioned into ...
1 vote
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### Does the cosine of a matrix have a geometric (non power series) interpretation? [closed]

You can adapt the power series definition of cosine to take in a matrix. Does this have a geometric interpretation/definition? Can it be used for various purposes? I actually have extended the matrix ...
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### Is there a conceptual reason why so many triplets of lines in a triangle are concurrent?

One of the striking phenomena one can't help but notice in elementary Euclidean geometry is how easy it appears to be to define triples of lines in a triangle which meet in a point. Now for each ...
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### Formula for "cointersection" of three circles?

I am working on the problem of finding "rational" dodecahedrons, and I have run across an interesting subproblem: How do you tell if three circles have a common intersection point? ...
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Question : Do you know this property of a hexagon? Consider the configuration: Six points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ in a plane and let six points $B_i \in A_iA_{i+1}$ for $i=1, 2,\dots, ... 19 votes 1 answer 799 views ### All saddles in the unit ball have area$<2\pi$? Let$M$be the saddle surface in$\mathbb R^3$defined by$x^2-y^2+z=0$. For any$r\geq 0$and$(x_0,y_0,z_0)\in\mathbb R^3$, let$rM+(x_0,y_0,z_0)$denotes the surface obtained by scaling$M$by$r$... 2 votes 1 answer 138 views ### Concyclic point made from Six arbitrary points Let$A_1A_2A_3A_4A_5$be irregular convex Pentagon and Let$P$be arbitrary point anywhere in Plane geometry. Let$X_1,X_2,X_3,X_4,X_5$be Circumcircle of$\triangle PA1A3$;$\triangle PA2A4$;$\... 7k views

### Is orientability a miracle?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$This question is prompted by a recent highly-upvoted question, Conceptual reason why the sign of a permutation is well-defined? The responses made ...
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### How big can a triangle be, whose sides are the perpendiculars to the sides of a triangle from the vertices of its Morley triangle?

Given any triangle $\varDelta$, the perpendiculars from the vertices of its (primary) Morley triangle to their respective (nearest) side of $\varDelta$ intersect in a triangle $\varDelta'$, which is ...
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### The outer Nagel points and unknown central circle

Na, Nb, Nc are the outer Nagel points. A'B'C' is the contact triangle. I claim that lines A'B', A'C', B'C' always cut the sides of the triangle NaNbNc at six points corresponding to an unknown circle. ...
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### Maximal number of vertices of the intersection of a flat and a hypercube

Consider the intersection of an $n$-dimensional hybercube and an $m$-dimensional flat (affine subspace) which contains the diagonal of the hypercube. This is a convex polytope. What is the maximal ...
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### Is the max-centre map continuous for open bounded domains?

Let $A$ be an open bounded subset of euclidean $n$-space $\mathbb{R}^n$. For $x\in A$, let $r=r(x)$ be the maximal radius such that the ball centred at $x$ with radius $r=r(x)$ is contained in $A$, i....
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### Two triangles have the same centroid theorem

Let $\triangle ABC$ and $\triangle A'B'C'$ be two triangles. The line through $A$ and perpendicular to $AA'$ meets the line through $B'$ and perpendicular to $BB'$ at $A_b$; The line through $A$ and ...