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.
561 questions
1 vote
2 answers
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Matrix formula for point of intersection of three circles through six points
Suppose we have the six points in the cartesian plane $(x_i, y_i)$ for $1 \leq i \leq 6$. Further suppose that we draw three circles through them, so that each circle passes through three of the ...
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3 votes
0 answers
59 views
Changes to the Delaunay Triangulation after deleting a point inside the convex hull
Consider the Delaunay Triangulation $\mathcal{DT}(P)$ of a finite set $P$ of points in the euclidean plane. let $CH\subset P$ be all points of $P$ that are on $P$'s convex hull, and not just the ...
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2 votes
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Relation of the most distant point-pair to the smallest enclosing circle
I am looking for a counter example to, resp. a proof of the correctness of, the following conjecture: among all pairs points from a finite set in the euclidean plane, that are at maximal distance, ...
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3 votes
0 answers
167 views
Are these two definitions of an NTA (nontangentially accessible) domain equivalent? If so, is the constant unchanged?
Jerison and Kenig give the following two conditions (alongside an exterior corkscrew condition, which is not relevant to this discussion) in the definition of an $(M,r_0)$ nontangentially accessible ...
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9 votes
0 answers
298 views
Twisted Rupert property
It has recently been proven that the 2017 conjecture that all convex polyhedra $P$ are Rupert is false: "A convex polyhedron without Rupert's property," Jakob Steininger, Sergey Yurkevich. ...
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11 votes
1 answer
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Outside-the-box in 3D: can the $27$ vertices of $\{0,1,2\}^3$ be visited with $13$ line segments connected at their endpoints, without repetition?
This is a variant of the 3D generalization of the well-known Thinking outside the box Nine dots problem I discussed in my previous MO post Optimal covering trails for every $k$-dimensional cubic ...
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3 votes
1 answer
480 views
Ptolemy theorem for spatial 4-gons
Given a closed spatial polygon with fixed edgelengths $r_i, i=1..4$ (that is a cyclicly ordered 4-tuple of vectors $v_i\in \mathbb R^3$ with $|v_i|=r_i$ such that $\sum v_i=0$) one can associate a ...
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0 votes
1 answer
107 views
Confusion regarding result of Hebbert on inscribed squares in quadrilaterals
The following concerns the 1914 paper The Inscribed and Circumscribed Squares of a Quadrilateral and Their Significance in Kinematic Geometry of Hebbert. Context. Hebbert presents THEOREM I. In every ...
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1 vote
1 answer
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Area of a triangle formed by parallel rays - A generalization of Zaslavsky's theorem
When I tried to find a special case of Dao's theorem on conics, I found the following result. I am looking for a proof of it. Let $ABC$ and $A'B'C'$ be two homothety triangle with $P$ is the ...
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2 votes
0 answers
141 views
Can a laser beam hit all points of $\{0,1,2\}^k \subset \mathbb{R}^k$ using $\frac{3^k-3}{2}$ mirrors only if emitted from outside the open $k$-cube?
Note: I already posted the $3$-dimensional version of this question on Mathematics Stack Exchange, but no answer has been received so far, so I hope that MathOverflow can be a more suitable place in ...
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3 votes
1 answer
255 views
A generalization of Newton–Gauss line
I am looking for a proof of a generalization of Newton–Gauss line as follows: Let ABC be a triangle, let a line $L$ meets $BC, CA, AB$ at three points $A', B', C'$ and let $A'', B'', C''$ be three ...
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3 votes
2 answers
214 views
The sum of the areas of the diagonal quadrilaterals in a quadrilateral grid
I am looking for a proof for the following result: Given a convex quadrilateral $ABCD$, divide each of its sides into $n$ segments of equal length (where n is an integer number). Then, connect the ...
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7 votes
3 answers
807 views
In Euclid's Elements, Book I, Proposition 47, Interpretation in terms of areas
Euclid's Elements, Book I, Proposition 47 is a statement and proof of the Pythagorean theorem, which involves the areas of squares adjacent to the three sides of a right triangle. In Euclid's Elements ...
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18 votes
2 answers
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Do the constructible lines and circles (not merely their intersections) cover the plane?
The universal compass and straightedge construction is the following. Start with two points in the plane—which we may as well think of as $0$ and $1$ on the horizontal axis—and iteratively carry out ...
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2 votes
2 answers
526 views
Sum of the angles of a triangle
Euclid's Proposition I.32 says that the sum of the internal angles of any triangle is equal to the sum of two right angles. However, there is a gap in Euclid's proof -- it does not show that the ray $...
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0 votes
1 answer
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Truth of a conjecture about the largest incircle of convex polygons
The problem of determining the largest incircle of a convex polygon can be solved by constructing the Voronoi diagram of the polygon's edges and selecting the vertex of the diagram's graph with ...
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2 votes
2 answers
337 views
Similar triangles and line in the plane
The construction of figures to produce special results in plane geometry is an interesting problem, attracting the attention not only of students but also of professional mathematicians. Pizza's ...
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3 votes
2 answers
679 views
With what probability does an inscribed/circumscribed triangle contain a point?
Three points uniformly selected on the unit circle form a triangle containing a point $R$ at distance $r \in [0;1]$ from its center with probability $P(r) = \frac{1}{4} - \frac{3}{2 \pi^2}\textrm{Li}...
17 votes
3 answers
2k views
How does Euclid know that two segments have a ratio?
In Elements (V, Def. 4), Euclid gives the following definition: Those magnitudes are said to have a ratio with respect to one another which, being multiplied, are capable of exceeding one another. ...
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2 votes
0 answers
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A special name for Hilbert's Axiom I.7?
The famous Hilbert's Axioms of Geometry include the Axiom I.7: If two planes have a common point, then they have another common point. Question 1. Was David Hilbert the first mathematician who ...
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10 votes
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Is there a double covering of the plane with lines using finitely many (but more than $2$) slopes?
This question is a follow-up to this question asked by Yaakov Baruch a few days ago. A double covering of the plane with lines is a set $\mathcal C$ of lines such that every point in the plane is ...
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18 votes
4 answers
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Double covering the plane with lines?
In Peter Winkler's Mathematical Puzzles (2024), it is shown that $\mathbb{R}^2$ can be double covered by a set of lines containing at least 3 different slopes (double covered = each point lies on ...
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-1 votes
1 answer
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Multitude centroids of hexagons are collinear
Using a computer, I found the following result. Now, I'm looking for a solution to prove this result: Let $A_{01}$, $A_{02}$, $A_{03}$, $A_{04}$, $A_{05}$, $A_{06}$ be six points in the plane, such ...
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11 votes
2 answers
868 views
Axiomatization of Euclidean geometry
The motivation of this question is finding an axiomatization of Euclidean geometry. I consider Tarski's axioms a satisfactory axiomatization of those parts of Euclidean geometry that do not include ...
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6 votes
1 answer
218 views
A Seven Circles Problem 2
I am looking for a proof a problem like Miquel's pentagram theorem as follows: Let $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ lie on a circle such that $A_1A_4$, $A_2A_5$, $A_3A_6$ are concurrent. Let $...
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5 votes
0 answers
149 views
A Seven Circles Problem
I am looking for a proof of the problem as follows: Let $A_1, A_2,\cdots, A_6$ lie on a circle with center $O$. Let the circle $(A_1OA_2)$ meets the circles $(A_3OA_4)$ again at $A_{23}$; the circle $(...
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2 votes
0 answers
166 views
Is it possible to solve the problem of doubling the cube using straightedge, compass, and the quadratrix?
"Quadratrix" (Wikipedia, 2024-11-14) writes: An accurate construction of the quadratrix would also allow the solution of two other classical problems known to be impossible with compass and ...
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1 vote
1 answer
189 views
Fundamental set of Euclidean lattices in higher dimensions
We consider the space of $n$-dimensional Euclidean lattices. Define the $i$-th successive minimum of a lattice $\Lambda$ as $ \lambda_i(L) := \min \left\{ r \in \mathbb{R}_{>0} \mid \dim_{\mathbb{R}...
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0 votes
0 answers
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Number of central $3$-dim hyperplane arrangements in generic position
Let $a(n)$ be the number of equivalence classes of $3$-dimensional central arrangements of $n$ hyperplanes in general position. I believe that $a(n)= 1$ for $n \leq 5$ and $a(6)=3$. Is this correct? ...
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0 votes
1 answer
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Expressing triangle area as a product of inradius and exradii: known result?
I encountered the following relationship while exploring triangle area computations. Given a triangle with inradius $r$, exradii $r_a, r_b, r_c$, and semiperimeter $s$, the area $\Delta$ satisfies: $$ ...
3 votes
0 answers
152 views
What is the status of Mahler’s open problems (from 1946) about the irreducibility of star bodies?
A duplicate of this: I am reading Mahler’s Lattice points in $n$-dimensional star bodies II (1946), in which he poses the following (then) open problems about the irreducibility of star bodies: Do ...
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0 votes
0 answers
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Coloring Kakeya's Needle
My question concerns a simple observation I made recently while working on the Perspective Three-Point Problem. I am wondering if anybody has a reference for the claim I am about to make concerning ...
1 vote
0 answers
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Solving special multivariable limits by Euclidean geometry
General Problem: Inspect function $L(a)$ for $a\in \mathbb{R}^{n}$, given that: $$L(a)=\lim_{x\to 0_{+}}\frac{x^{a}}{F(x)}$$ Notation legends: $x=(x_1,\ldots,x_n),\ a=(a_1,...,a_n),\ x^a=x_{1}^{a_1}\...
user500489
2 votes
1 answer
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Does this result above six points follow have a name?
Does this result above six points follow have a name? Let $A$, $B$, $C$, $D$, $E$, $F$ be six points in the plane and $AB, CF, ED$ are concurrent and $BC, DA, FE$ are concurrent then $CD, EB, AF$ are ...
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3 votes
0 answers
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Pólya's orchard problem among Gaussian primes
Quoting myself from an earlier post: Pólya's orchard problem asks for which radius $r$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of ...
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0 votes
0 answers
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Sum of Simplex Volumes with Corners from Points in Convex Configuration
Question: given $k,\,k>n$ points in convex configuration and general position in $n$ dimensional Euclidean space, i.e. no $n+1$ points of which are co-hyperplanar, what can be said about how the ...
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2 votes
1 answer
282 views
How can one test whether a given analytic curve in the plane is algebraic or not?
Suppose $\Gamma$ is an analytic Jordan curve in the complex plane $\mathbb{C}$. What are ways to test whether $\Gamma$ is (contained in) an algebraic curve, i.e., whether there exists a real ...
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2 votes
0 answers
103 views
Theta series of well-rounded lattices
I've started looking into well-rounded Euclidean lattices and I was interested in learning whether their theta series have any interesting properties, but haven't found much in terms of bibliography ...
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12 votes
5 answers
1k views
Dissection proof of Heron's formula?
In his recent book, Love Triangle, Matt Parker playfully complains that Heron's formula is an "opaque formula, and I feel like you just chuck in the side-lengths, turn a series of arbitrary ...
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32 votes
2 answers
3k views
Is every connected subgroup of a Euclidean space closed?
The question listed above (in the context of the complex numbers, but it is a reasonable question to ask in any dimension) was asked by a student in my complex analysis class, and I did not have an ...
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0 votes
0 answers
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Upper bounds for minimum angle
What are the latest and best results on the asymptotic upper bound for the minimum angle between any pair of rays among $n$ rays in $\mathbb{R}^3$? Any helpful answer would be appreciated. Thank you!
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2 votes
2 answers
502 views
A necessary and sufficient condition for three diagonals of a hexagon to be concurrent
When talking about the condition for the three diagonals of a hexagon to be concurrent, we will think of Brianchon's theorem. Using software, I discovered a necessary and sufficient trigonometric ...
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5 votes
1 answer
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Is this a new result about hexagon?
Let a hexagon $AB'CA'BC'$ let $AB' \cap A'B=C''$, $BC' \cap B'C = A''$, $CA' \cap C'A = B''$ then three conditions as follows equivalent: Three lines $AA', BB', CC'$ are concurrent (let the point of ...
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2 votes
0 answers
150 views
Another Butterfly theorem — Conway like circle
Have You seen these result as follows before? In Figure 1: $AA'=BB'=tAB$; $CC'=DD'=tCD$, where t is real number then $ABCD$ is a cyclic quadrilateral iff $A'B'C'D'$ is a cyclic quadrilateral. In the ...
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2 votes
1 answer
237 views
Incenter-of-mass of a polygon
"Circumcenter of mass" is a natural generalization of circumcenter to non-cyclic polygons. CCM(P) can be defined as the weighted average of the circumcenters of the triangles in any ...
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3 votes
0 answers
564 views
What's the number of facets of a $d$-dimensional cyclic polytope?
A face of a convex polytope $P$ is defined as $P$ itself, or a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...
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10 votes
1 answer
592 views
If we know the combinatorics of a polyhedron, and all but one of its dihedral angles, does that uniquely determine the remaining dihedral angle?
If we know the combinatorics of a polyhedron, and all but one of its dihedral angles, does that uniquely determine the remaining dihedral angle? I’m happy to assume the polyhedron is simply connected, ...
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3 votes
2 answers
291 views
Triangle centers formed a rectangle associated with a convex cyclic quadrilateral
Similarly Japanese theorem for cyclic quadrilaterals, Napoleon theorem, Thébault's theorem, I found a result as follows and I am looking for a proof that: Let $ABCD$ be a convex cyclic quadrilateral. ...
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9 votes
1 answer
651 views
Does the sequence formed by Intersecting angle bisector in a pentagon converge?
I asked this question on MSE here. Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $...
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10 votes
0 answers
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What axiomatic system does AlphaGeometry use?
In January 2024, researchers from DeepMind announced AlphaGeometry, a software able to solve geometry problems from the International Mathematical Olympiad using a combination of AI techniques and a ...
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