geometry assuming the parallel postulate of Euclid: in a plane, given a line and a point not on that line, there is exactly one line parallel to the given line through the given point.

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Rectangular problem

I was trying to solve this problem: Let P be a point in the interior of rectangle ABCD. Given PA = 3, PD = 4 and PC = 5, find PB. I feel lost because it's not right to assume P is in the center ...
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1answer
971 views

How to plot N points on the surface of a D-dimensional sphere roughly equidistant apart?

Let's say I have a D-dimensional sphere with a radius R. I want to plot N number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter where ...
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1answer
108 views

Finding the x- coordinate in triangle

Is it possible to find the point which is marked by question mark ? we know that the s1(x)=s2(x) (the areas of the two triangles are equal)
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473 views

Cutting the corners of a cube

Do you know of any way to cut the corners of a cube by means of rotation assuming that the cube is centered in the origin of XoYZ? For example if we have a square centered in the origin of XoY and we ...
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2answers
222 views

Intersection of a unit sphere of a given norm in finite dimension with an hyperplane.

Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$. Let $C:=\{x\in\mathbb{R}^n\,:\,\|x\| \leq 1\}$, that is to say let $C$ be a convex compact symmetric set of non empty interior. Let $H$ be a linear ...
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1answer
79 views

Locus perpendicular to a plane in $\mathcal{R}^4$

I have solved an exercise but I'm not sure to have solved it perfectly. Could you check it? It's very important for me.. In $\mathcal{R}^4$ I have a plane $\pi$ and a point P. I have to find the ...
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1answer
652 views

find the center of an ellipse given tangent point and angle

I have an ellipse with known major radius $r_x$ and minor radius $r_y$, aligned with the x- and y-axis. Given a tangent point $T$ and the tangent angle $\alpha$, how do I calculate the center $C$ ...
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1answer
68 views

find point on the line in $R^n$

I am trying to find the coordinates of a middle point of a line in $\mathbb{R}^n$. Let $X(x_1, \ldots, x_n)$ and $Y=(y_1, \ldots, y_n)$ be two points in $\mathbb{R}^n$. How do I find the middle ...
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2answers
2k views

Solid angle between vectors in n-dimensional space

There is a formula of to calculate the angle between two normalized vectors: $$\alpha=\arccos \frac {\vec{a} \cdot\ \vec{b}} {|\vec {a}||\vec {b}|}.$$ The formula of 3D solid angle between three ...
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0answers
91 views

What property does this equation calculate?

It's pretty difficult to Google for the meaning of a formula. This is the equation, it has to do with ellipses and GIS coordinates. $$\nu =\frac{ a} {\sqrt{(1 - (e^2 \cdot \sin(\varphi))^2)}}$$ $a$ ...
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1answer
117 views

Equivalence to the Euclidean Parallel Postulate

Show that the proposition P : There exists a pair of straight lines that are at constant distance from each other. is equivalent to the Parallel Postulate Q : If two lines are drawn which ...
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75 views

Partition of open sets in $\mathbb{R}^d$.

Let $\Omega\subset\mathbb{R}^d$ be open. We want to find a good partition of $\Omega$ into more elementary sets. In particular we want compact sets $K_j$'s and open sets $V_j$'s such that ...
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1answer
246 views

line is the perpendicular bisector of a segment obtained by reflecting twice through two rays

Given two rays, L and M, with common origin O, and a point Q inside the acute angle formed by the rays, reflect Q across L to obtain Q' and then reflect Q' across M to obtain Q''. Similarly, reflect ...
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3answers
2k views

At what coordinates should the fourth vertex be located?

A parallelogram is drawn on a coordinate grid so that three vertices are located at $(3, 4)$, $(-2, 4)$ and $(-4, 1)$. At what coordinates should the fourth vertex be located?
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1answer
226 views

Length bisection from circular arc

I am not sure if the following result is well known. I stumbled across it from the paper The Perimetric Bisection of Triangles by Dov Avishalom, where the result was stated without proof. I am ...
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3answers
422 views

Euclidean geometry exercise

I would like some help to solve this: Consider a triangle $\triangle ABC$ with $\angle A$ a right angle and $BC=20$. Divide $BC$ into four congruent segments, that is, take the points ...
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2answers
75 views

Simple Vector Question in $\mathbb{R}^3$

Two points $A$ and $B$ in $\mathbb{R}^3$ with origin $O$ are given in terms of a Cartesian coordinate system by $A = (1, 2, 3)$ and $B = (4, 5, −1)$. How do you find the point $C$, such that $OACB$ ...
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1answer
124 views

Is there a geometric proof to answers about the 3 classical problems?

I know that there is a solution to this topic using algebra (for example, this post). But I would like to know if there is a geometric proof to show this impossibility. Thanks.
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2answers
402 views

Showing that an Isometry on the Euclidean Plane fixing the origin is Linear

Suppose $f$ is an isometric (i.e., distance preserving) function on $\mathbb{E}^2$ such that $f(0,0) = (0,0)$. Then I want to show that $f$ is necessarily linear. Now $f$ is linear iff $f$ is both ...
3
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0answers
229 views

Ellipse and circles

Playing with an ellipse I discovered the following properties and am now looking for a nice proof or references. Let's consider an ellipse with foci $A$ and $B$ and let $C$ be a point on it. Let $l$ ...
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1answer
93 views

Is the sum of the diagonals of an isosceles trapezoid at least the sum of the bases?

Here's one question that has been bothering me now for a while. It is not homework. For an isosceles trapezoid (wikipedia link: http://en.wikipedia.org/wiki/Isosceles_trapezoid), do we always have ...
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1answer
987 views

Relevance of Euclidean geometry to modern mathematics

I'm interested whether, barring the reasons such as the importance of historical narrative and having an illustrative example, one should learn (synthetic) Euclidean geometry. To make such an inquiry ...
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1answer
923 views

calculating volume of a horizontal cylindrical tank from depth

Has anyone found a good formula to convert a depth of fluid to a volume remaining in a tank for a cylindrical tank laying horizontal? (with or without half sphere end caps) Much Thanks!
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1answer
72 views

Notation-Linear Algebra/ Euclidean Geometry

This is a notational question: What does $\overset{\underset{\mathrm{\Delta }}{}}{=}$ denote in $-0.5JDJ\overset{\underset{\mathrm{\Delta }}{}}{=} X^TX$ where $J=I-n^{-1}ee^T$ with $e$ being a ...
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3answers
1k views

How to prove that the exterior angle sum of an n-sided polygon is 360 degrees?

Just wondering about the smartest and best way to prove such a question. I know of many ways, and also I don't want to use anything related to $180(n - 2)$.
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1answer
456 views

Existence of minimum distance between two closed sets, one of which is bounded

I would like to prove the following statement: Let $S,T \subseteq \mathbb{R}^{n}$ be closed sets with $S \cap T = \emptyset$, at least one of which is bounded. Then there exist $x \in S$ and $y \in T$ ...
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1answer
320 views

Proof of Archimedes Lemma about the Center of Mass

I am looking for a proof of the following statement, known as Archimedes' Lemma: If an object is divided into two smaller objects, the center of mass of the compound object lies on the line segment ...
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1answer
432 views

5 Points uniformly placed on a sphere

I have a sphere and I have to place some points on it, the most uniformly possible. If I have 4 points, placing them as vertices of a tetrahedron seems good. If I have 6 points, placing them as ...
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1answer
914 views

Maximize the area of the inscribed triangle

Problem Try to determine the maximum area of the inscribed equilateral triangle of a ellipse with semi-major axis $a$ and semi-minor axis $b$. Thoughts Suppose the equilateral triangle is ...
2
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0answers
633 views

Equations of branches of a mind map

Sorry for the long question, but it's not so simple to explain. Consider a mind map like this: I want to draw branches in a cartesian coordinate system. I'd like to find two equations which ...
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1answer
114 views

Base of shortest connection of two skew straight lines

During tutoring (12th grade, regular Math class), I had to explain how to find the two points $s$ and $s'$ that are the base of the perpendicular connection between two skew straight lines $g$ and ...
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2answers
138 views

Probability that two infinite cylinders overlap

Given two infinite cylinders: radius $r_0$, centered at the origin and pointing $\hat u_1=(0,0,1)$ radius $r_1$, centered at $(d,0,0)$ and pointing at $u_2$ For a given distance $d$, if $u_2$ is ...
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0answers
159 views

Apollonius' circle

A theorem states, that the three Apollonian circles, associated with the given triangle $ABC$ with sidelengths $a \neq b \neq c$ intersect in two points. The proof proceeds by showing that if the two ...
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1answer
326 views

Intersection of Two Simplices

How to find vertices a the polytope-intersection of two simplices, if I know the vertices of these simplices. More precisely: Let $T_1$ and $T_2$ be two regular $n-1$ dimensional simplices with ...
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0answers
57 views

Morphisms of Euclidean Geometry

In Euclidean geometry $\mathbb{E}^2(\cong \mathbb{C})$, the group $G=\{z\mapsto raz+b, z\mapsto ra\bar{z}+b\colon a,b\in \mathbb{C},|a|=1, r\in \mathbb{R}^{+} \}$ is precisely the group of ...
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3answers
5k views

The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?

On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true?
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1answer
102 views

A question about a metric space

I have a question regarding the following problem. Let $A,B,C$ be the three independently selected, uniformly distributed points on the unit sphere $S^3$ in $\Bbb R^4$. What's the probability ...
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3answers
181 views

How many unique distances are there in a 5 x 5 grid?

I cannot figure this out: I have a square in the plane with side length $5$. $A$ and $B$ are points in the square. The coordinates of $A$ and $B$ are always integers. I want to know how many unique ...
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2answers
197 views

Higher-dimensional Extension of Triangle Geometry?

I am currently exploring generalizations of triangle geometry to higher dimensions. I know that "important questions" of Euclidean geometry have been already addressed and is considered obsolete; most ...
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2answers
116 views

Regular Polyhedrons

In $\mathbb{R}^3$, there are five regular polyhedrons (up to similarity), and can be parametrized by number of vertices, edges and faces. What is the number of regular polyhedrons in $\mathbb{R}^n$, ...
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1answer
61 views

In an N-dimensional space filled with points, systematically find the point with highest spearmans correlation to a given-point

I asked a question exactly like this a while ago, so I do not know if it is appropriate to ask pretty much the same question with a single tweak. For the record, my first question is In an ...
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1answer
195 views

In an N-dimensional space filled with points, systematically find the closest point to a specified point

This is my first post on the mathematics stack exchange site. If I am posting this question on the wrong site, I apologize and please let me know so I can delete it. I am a programmer, and one thing ...
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0answers
231 views

Evaluating the average distance from a point in the unit disk to the disk

I am interested in finding the average Euclidean distance from a point $(x,y)\in\mathbb{D}_2$, the unit disk $\{(u,v):u^2+v^2\leq 1\}\subseteq\mathbb{R}^2$, to the disk $\mathbb{D}_2$. This amounts to ...
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2answers
257 views

Thales' theorem and point in circle

Can someone explain to me why $\alpha' < \alpha$ and $\beta'<\beta$ when point $p_l$ is inside the circle? There is suppose to be a way to see this using Thales' theorem. Also if $p_l$ is ...
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1answer
194 views

A curve that intersects every plane in finitely but arbitrarily many points

Does there exist a piecewise smooth curve in $\mathbb{R}^3$ such that every plane intersects the curve at finitely many points and the number of intersection points can be arbitrary large? If the ...
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1answer
122 views

Intersection of 2 $p$-simplices is a finite union of some $p$-simplices

I'm looking for a non-painful proof of this assertion. A p-simplex is defined as the set of all sums $\sum_{i=0}^p t_i x_i$ with $0\leq t_i\leq 1$, $\sum_{i=0}^p t_i=1$ for a geometrically ...
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2answers
260 views

This classic from euclid's elements, is it accepted everywhere?

I was reading linear vector spaces. When doing some exercise to prove some statements based on the properties defined for linear vector spaces, i suddenly noticed, outside the things defined, i'm ...
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3answers
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Why is the inradius of any triangle at most half its circumradius?

Is there any geometrically simple reason why the inradius of a triangle should be at most half its circumradius? I end up wanting the fact for this answer. I know of two proofs of this fact. Proof ...
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1answer
371 views

Distance Formula for n-dim Barycentric Coordinates

Assume that we are given every distance between each pair of points from a $n$-simplex $\triangle$. Given $n$-dimensional barycentric coordinates (measured with respect to $\triangle$) of two points, ...
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1answer
62 views

Solving vector equation

Suppose I know the following 3 normalized vectors in 3-D Euclidean space: $\frac{\vec{A}}{||\vec{A}||}$, $\frac{ \vec{A} + \vec{B}}{||\vec{A} + \vec{B}||}$ $\frac{ \vec{A} + \vec{C}}{||\vec{A} + ...