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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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 $\triangle PA1A3$; $\triangle PA2A4$; $\...
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57 votes
4 answers
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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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4 votes
1 answer
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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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4 votes
1 answer
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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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5 votes
0 answers
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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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2 votes
1 answer
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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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An equivalent characterization of conical surfaces

Consider an $n-1$ dimensional surface in $\mathbb{R}^n$. If the tangent plane at any point of this surface always passes through the origin, can we show that the surface must be a conical surface? 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 ...
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21 votes
3 answers
1k views

Are there infinitely many "generalized triangle vertices"?

Briefly, I'd like to know whether there are infinitely many "generalized triangle centers" which - like the orthocenter - are indistinguishable from a vertex of the original triangle. This ...
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1 vote
0 answers
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Proving convexity of total distance between two parties with one meeting point [closed]

Apologies if I´m not using the correct mathematical wording or notation. I´ll try my best in the following problem Suppose you have a coordinate system and 5 coordinates ($A1$, $A2$, $B1$, $B2$, $M_{x,...
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Maximizing the area of a region involving triangles

I thought of a question while making up an exercise sheet for high school students, and posted it on MathStackExchange but did not receive an answer (the original post is here), so I thought perhaps ...
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13 votes
1 answer
267 views

A randomized version of straight-edge and compass construction

Suppose you start with two points in the plane which are distance 1 apart, which for concreteness can be $(0,0)$ and $(0,1)$. Then you keep marking new points based on ruler and compass constructions. ...
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1 vote
0 answers
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Pseudo-Droz-Farny circles

I would like to present a construction of 2 circles. These 2 circles are somewhat similar in appearance to the well known Droz-Farny circles that can be drawn for every isogonal conjugate pairs of ...
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1 vote
0 answers
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Definition of a unit ball in an Euclidean subspace? [closed]

Suppose $\Lambda$ is a $3$ dimensional lattice inside $\mathbb{R}^4$ and let $E$ be the subspace $\mathbb{R}$-spanned by $\Lambda$. What exactly is meant by the unit ball in $E$? This is something ...
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5 votes
1 answer
221 views

Lemoine-Lozada circles

I made some rookie attempt to define the 4th Lemoine circle recently. The alternative name for this circle was suggested yesterday. Further investigation revealed a family of circles associated with ...
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1 vote
0 answers
103 views

How do Euclid’s postulates imply that a line has more than two points? [closed]

How do Euclid’s postulates imply that there exist more points than provided as assumptions, e.g. in the statement: Let C1 be the circle centered at A, with radius AB, let C2 be the circle centered at ...
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2 votes
0 answers
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Historical question about tangent lines to disjoint circles

It is pretty well known that two disjoint circles have 4 different lines that are simultaneously tangent to both circles. There are constructions with ruler and compass available in many books, but I ...
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12 votes
1 answer
7k views

The 4th vertex of a triangle?

I was immensely surprised and amused by the idea of the fourth side of a triangle that was introduced by B.F.Sherman in 1993. 'Sherman's Fourth Side of a Triangle' by Paul Yiu is available here. ...
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2 votes
1 answer
213 views

Constructing an equilateral triangle using Tarski's axioms for geometry

In Euclid's first geometry proposition, he constructs an equilateral triangle given an arbitrary line segment. I was wondering if it was possible to prove this straight from Tarski's axioms for ...
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5 votes
2 answers
345 views

The 4th Lemoine circle

The first and second Lemoine circles are well-known to geometers. According to this article the third Lemoine circle has been first discovered by Jean-Pierre Ehrmann in 2002. It is worth noting that ...
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14 votes
2 answers
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Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?

I've posted this question on Math Stack Exchange, but I want to bring it here too, because 1) the proof seems missing in the literature, although they are some sporadic mentions and 2) maybe it ...
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1 vote
1 answer
63 views

Collinearity in tangential pentagon [closed]

I am looking for a proof of the following claim: Given tangential pentagon. Touching point of the incircle and the side of the pentagon,the vertex opposite to that side and the intersection point of ...
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5 votes
1 answer
418 views

Is there a mapping from Euclidean space to real numbers which relatively preserves distance? [closed]

Motivation: I need to find a mapping from $n$-dimensional Euclidean space to real numbers such that the distance between each pair of points in the quoted space is relatively-preserved after the ...
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3 votes
1 answer
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The product of the lengths of two line segments that belong to Newton line [closed]

I am looking for the proof of the following claim: Consider a family of bicentric quadrilaterals with the same inradius length and the same distance between incenter and circumcenter. Denote by $P$ ...
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-5 votes
1 answer
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Is this a valid definition of Euclidean geometry? [closed]

In trying to understand what actually constitutes a "geometry" I came across many definitions of Euclidean spaces and geometries. Euclidean space is defined as an affine space with an inner ...
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4 votes
2 answers
179 views

Six conelliptic points

Can you prove the following proposition: Proposition. Given an arbitrary triangle $\triangle ABC$. Let $D,E,F$ be the points on the sides $AB$,$BC$ and $AC$ respectively , such that $\frac{AB}{DA}=\...
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3 votes
1 answer
728 views

Three circles meet at a point [closed]

I am looking for the proof of the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with circumcenter $O$. Let $A',B',C'$ be a reflection points of the points $A,B,C$ ...
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2 votes
0 answers
242 views

A problem on configuration of Dao's theorem on six circumcenters

Abstract: In the figure belows: Three lines through center of pair opposite red circle are concurrent. This is a statement of Dao's theorem on six circumcenter, a new theorem in plane geometry which ...
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5 votes
0 answers
219 views

Arrangement of points, lines, and planes

Is it possible to construct a finite nontrivial arrangement of points, lines, and planes in 3-dimensional Euclidean space with the following properties? every line is incident with four points and ...
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2 votes
0 answers
61 views

Principal diagonals of octagon meet in a single point

Can you provide a proof for the following claim: Claim. Given octagon circumscribed about an ellipse. If the vertices of the octagon lie on another ellipse then its principal diagonals meet in a ...
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1 vote
1 answer
66 views

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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13 votes
2 answers
2k views

Is it a new discovery on conic section?

I discovered a problem in plane geometry (there are some nice special cases) as follows: Let $ABC$ be a triangle and $\Omega$ be arbitrary circumconic. Let two points $A_b, A_c \in BC$, $B_c, B_a \in ...
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1 vote
1 answer
106 views

Generalizing Bottema's theorem

Can you provide another proof for the claim given below? Claim. In any triangle $\triangle ABC$ construct triangles $\triangle ACE$ and $\triangle BDC$ on sides $AC$ and $BC$ such that $\frac{AE}{AC}...
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1 vote
1 answer
317 views

Cramer–Castillon problem like

Special case of Golden ratio as a property of conic section (is it known?) as follows: Let $ABC$ be arbitrary triangle and $DEF$ is the its tangential triangle. Let $CF$ meets $AB$ at $G$ and $BE$ ...
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3 votes
1 answer
88 views

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 ...
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  • 2,615
1 vote
0 answers
42 views

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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4 votes
0 answers
155 views

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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6 votes
0 answers
339 views

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 ...
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18 votes
2 answers
522 views

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 ...
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6 votes
4 answers
396 views

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 $\triangle ABD$,$\triangle ABC$,$\triangle BCD$ ...
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  • 2,615
7 votes
3 answers
1k 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 ...
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  • 2,615
2 votes
1 answer
73 views

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 ...
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  • 2,615
-9 votes
1 answer
164 views

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 ...
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45 votes
2 answers
3k views

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 ...
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8 votes
1 answer
236 views

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 ...
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10 votes
1 answer
157 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 $K$-...
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4 votes
1 answer
181 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 ...
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  • 2,615
5 votes
0 answers
261 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 ...
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6 votes
1 answer
183 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 ...
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  • 2,615
5 votes
1 answer
240 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 ...
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