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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Solving inverse triangle-center problems
I came across this problem by trying to construct a "nice" triangle, e.g. for an illustration, from 3 distances $\lbrace\|A-O\|, \|B-O\|, \|C-O\|\rbrace$ and the fuzzy requirement that $O$ ...
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Fast calculation of the circum hyperspheres of n-simplices
first the trivial facts:
Non-degenerate n-dimensional simplices have $n+1$ corners and $\frac{(n+1)n}{2}$ edges.
The center of the circum-hypersphere from which all $n+1$ corners are equidistant can ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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