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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.

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.

501 questions
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A problem in elementary geometry

Let us have a triangle ABC in the Cartesian plane and consider the following transformation of this triangle: On the ray AB starting at A, select a point B' so that so that |AB'|=|AC|. Likewise, on the ray BC starting at B, select a point C'…
Deepti
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ten concurrent lines

this relates to a question asked at MSE. i was able to find an answer using complex numbers. here is the question: there are five points on a circle. take any three points, through the centroid of the three points draw a line orthogonal to the line…
abel
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Law of sines for tetrahedra

Wikipedia gives a generalization of the law of sines to higher dimensions, as defined in F. Eriksson, The law of sines for tetrahedra and n-simplices. However, this generalization misses an important point about the standard law of sines, which…
Craig
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Inter-Kissing Number for Non-Spheres

In 3D, the maximum number of spheres which can inter-touch is $5$ (MO question Inter-Kissing Number for Spheres of Different Sizes). This maximum reduces to $4$ for unit spheres. Is there a different shape (e.g., an egg, or a pyramid) for which…
bobuhito
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Generalising Euler's formula to ellipses and three dimensions

Let $D$ be the closed unit disk, $T$ a triangle and $E$ an ellipse with $E\subset T \subset D$. Without loss of generality say that $E$ is centred at Cartesian coordinates $(c, 0)$ with $0\leq c \leq 1$ being a variable parameter. The axes of $E$…
Antony
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Conjectute: no exist an equilateral triangle such that all vertices are integer numbers

I am looking for a solution for a conjecture as follows. In Cartesian plane, no exist an equilateral triangle such that three vertices are integer numbers. I hope that you like the question and let me a answer.
Oai Thanh Đào
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Orthogonality between vectors whose components increase

This is a cross-posting of a MSE question (which did not receive any feedback there so far). Say that a vector $x=(x_1,x_2, \ldots ,x_n)\in {\mathbb R}^n$ is nondecreasing if $x_1 \leq x_2\leq \ldots \leq x_n$. Can anyone show or find a…
Ewan Delanoy
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Why 2 as an exponent in the euclidean vector space?

Let us develop the question: Let us focus on finite real vector space, equiped with a norm. A priori, one does not make the hypothesis that the norm is derived from a scalar product. Which structural condition could be asked on such space that is…
Lucas
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Estimate on the minimum distance from integer points on some fixed hyperplanes to a moving hyperplane

Suppose in $\mathbf{R}^n$ there are $m$ given hyperplanes $\Pi_j:\sum_{i=1}^n c_{i,j}e_i=0$ all of which go through the origin, and all the coefficients $c_{i,j}$ are rational (you can make them all integers if you like). Clearly, each hyperplane…
Haoran Chen
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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. 1/2 mm) using these tools. Assume I have raw…
user3653831
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How do Euclid’s postulates imply that a line has more than two points?

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 B, with radius BA. let F be a point of…
Shillington
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ratio between a polygon bounded in another polygon

Let A be a convex polygon with area SA. Construct a new polygon B by orderly connecting the midpoints of the segments of A. Denote the area of B by SB. Claim : the ratio SB/SA is constant for all polygons of the same number of vertices. This ratio…
Arie Rokach
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Euclid Book 1 Proposition 4

In Euclid's The Elements, Book 1, Proposition 4, he makes the assumption that one can create an angle between two lines and then construct the same angle from two different lines. I do not see anywhere in the list of definitions, common notions, or…
Jessica
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finding subsuming hypervolumes

Imagine we have an N-dimensional space where each dimension can only have integer values. Imagine further that this space has a set of hypercubes scattered about, each hypercube with its own position and dimensions. How can we compile a decision…
Mark Klein
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About the area of the region where the paper is twofold when you double a piece of paper in the shape of a triangle

Suppose that you have a piece of paper in the shape of a triangle $ABC$ whose area is $S_0$ and that the area of the region where the paper is twofold when you double the paper in two along a line is represented as $S$. Then, supposing that $BC=a\ge…
mathlove
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