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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locus of a variable straight line [on hold]

Geometry: A variable straight line always intersects the lines x=c,y=0; y=c,z=0; z=c,x=0. find the equation to its locus. taking the equation of a line in parametric form and substitute the given ...
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41 views

Why are carrom boards square? [on hold]

This question may seem a little off-topic for this site.... We have all seen carrom boards.Now,why are carrom boards always square and not rectangular?Is it only because distance to the pockets will ...
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1answer
18 views

Finding $y_{A,i}$ and $y_{B,i}$ in this geometric relationship problem.

Finding $y_{A,i}$ and $y_{B,i}$ in this geometric relationship problem. I'm an high-speed aerodynamics student. I am studying a sweptback wing like in the figure below (in green). Notice that I ...
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2answers
37 views

Probably very basic Euclidean geometry; Why is the following expression valid for a point along a straight line?

I am looking at constructible points in abstract algebra, particularly in $\mathbb{C}$. Alongside a proof of a theorem, I came across this expression which I cannot work out how it's been derived. It ...
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1answer
25 views

Four Spheres Intersect Along Circles: Prove That Circles' Planes Are Either $\parallel$ to The Same Line, Or Have a Common Point

Problem: Let $\,A,\,B,\,C,\,D\,$ be four distinct spheres in a space. Suppose the spheres $A$ and $B$ intersect along a circle which belongs to some plane $P$, the spheres $B$ and $C$ intersect ...
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Finding closest vector for all rows in a matrix

I have two matrices 1. D ($m \times n$) and 2. C ($k \times n$). Typically, $m \approx 10^4, n \approx 100, k \approx 100 $. For each row r in D, I need to find the index of the row in C that's ...
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1answer
29 views

Partitioning a convex object without cutting existing convex subsets

$C$ is a convex object in the plane. $D_1,\dots,D_n$ are pairwise-disjoint convex subsets of $C$ such that $D_1\cup\dots\cup D_n \subsetneq C$, like this: Is it always possible to partition $C$ to ...
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1answer
47 views

Size of a point. [duplicate]

I know this may sound too simple or maybe too absurd to discuss, but I am having a hard time visualizing a point in space! In Euclid's Elements a 'Point' is defined as Something which has no part. ...
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19 views

A vector which is perpendicular to two vectors not in the same plane

Assume that I have two vectors $v_1, v_2$ which are not parallel and they don't lie the same plane. How to find a third vector $n$ perpendicular to $v_1$ and $v_2$? You could take the cross prosuct, ...
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1answer
24 views

Conjugating rotation by another rotation

If $g ∈ \mathrm{SO}(3)$ is the rotation about axis $p$ by angle $α$, and $h$ is a rotation mapping $p$ to another line $q$, then $g$ conjugated by $h$ is the rotation about $q$ by the same angle $α$. ...
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1answer
15 views

How to rotate in quaternions but for 2d version for arbitrary angle?

I am trying to understand the idea behind rotating in quaternions, but first I want to understand the math for 2d rotation. I saw some youtube videos, and I know that for 2D, a point in 2D can be ...
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1answer
27 views

Is the Probability of Selecting 3 Random and Colinear Points nil?

Recently, the mathematics YouTube channel released a video titled "Triangles have a Magic Highway - Numberphile". In the video, at 6:40, the expert being videoed says that the probability of any three ...
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22 views

A right hexagon and right pyramid

Does it possible to obtain a regular hexagon as a section of right pyramid with the base of the form of regular pentagon? O.Ganyushkin
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1answer
10 views

How to prove that the pointreflection at the midpoint of two several points out of a regular pointlattice fix the lattice?

How to prove that the pointreflection at the midpoint of two several points $A,B\in\mathfrak{L}$ in a regular pointlattice $\mathfrak{L}$ fix the lattice $\mathfrak{L}$? We call ...
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1answer
37 views

Equivalence between constant and positive metric and usual $\Re^3$ metric

I'm trying to answer the following question: Is any positive and constant metric in $\Re³$ equivalent to the usal metric defined as $$ds² = dx² + dy² + dz² \tag{*}\label{1} $$ with $ds = ...
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0answers
26 views

What does it take to have a precise definition of volume?

Many proofs in elementary geometry use an intuitive but imprecise definition of the area or the volume. For example, Euclid's first proof of the Pythagorean Theorem uses the fact that all triangles of ...
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18 views

Notation for Line Segment vs. Directed Line Segment

This may be nit-picky, but I noticed inconsistencies in a high school math text I was reading, and I'm curious what the world thinks. For the most part throughout this textbook, notation is used as ...
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2answers
119 views

Prove $(a^2+b^2+c^2)(a+b-c)(b+c-a)(a+c-b)\leq abc(ab+bc+ac)$

Let $a,b,c$ are $3$ edge of a triangle. Prove $(a^2+b^2+c^2)(a+b-c)(b+c-a)(a+c-b)\leq abc(ab+bc+ac)$. My try: I suppose $c=\min\{a,b,c\}$ but I don't know what next.
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1answer
45 views

What is the equation of a pyramid with a square base?

Which algebraic description can be found for a pyramid, defined as a scalar function $$f:\mathbb{R}^2 \rightarrow \mathbb{R}$$ $$(x,y)\rightarrow z$$ Particular assumptions: Square base $z=0 \iff ...
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2answers
67 views

Proof that obtuse angles = 90 degrees

There's a proof on how every obtuse angle is equal to 90 degrees, and I can't seem to find the issue. Given: Quadrilateral ABCD, AD=BC, ∠ADB is obtuse, m∠CBD=90 Drawing ...
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31 views

Average distance between nearest neighbors for randomly placed points in a unit square?

The answers I found were generally about the distance between any two points in a square. I'm trying to find the average distance between nearest neighbors. Background on this is I'm processing 3D ...
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1answer
27 views

Quantify how similar a list of four numbers is.

I'm working on a program in which a user generates four distinct values from 1-256. I'd like to compare these four user generated values to two pre-generated lists of values and determine which of the ...
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2answers
84 views

Finding segment in a right triangle.

Here is the picture of the question: $ABC$ is a right triangle. $m(CBA)=90^\circ$. $m(BAD)=2m(DAC)=2\alpha$. $D$ is a midpoint of $[BC]$. $E$ is a point on $[AD]$. ...
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1answer
39 views

$20-80-80$ triangle, rhombus with orthocenter, circumcenter

Let $ABC$ triangle such that $\angle A=20^{\circ}$ and $\angle B=\angle C=80^{\circ}$.Let $D,E$ be point on lines $AC,AB$ respectively such that $BD,CE$ are angels bisector of triangle $ABC$.Let $H,O$ ...
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2answers
29 views

Height of a paralelogramm

I have the coordinates of the 4 vertexes of a parallelogram. If i calculate the length of two opposing sides, how do I get the perpendicular distance between them? Is it just the distance between the ...
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2answers
42 views

prove every pair of points $P,Q, d(P,Q)>0$ [closed]

Prove: For every pair of points $P, Q$ 1. $d(P,Q)>0$ 2. $d(P,Q) = 0$ if and only if $P=Q$ 3. $d(P,Q) = d(Q,P)$ where $d(P,Q)$ is defined as the distance between $P$ and $Q$ and $d$ is a function ...
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1answer
28 views

Locus of vertex

A variable parabola of latus rectum $l $, touches a fixed equal parabola , the axes of the two curves being parallel . Then locus of the vertex of the moving curve is a parabola , then what is the ...
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1answer
27 views

Point of intersection of tangents

If the distance of two points $P$ and $Q$ from the focus of of a parabola $y^2 =4ax$ are $4$ & $9$ then what is the distance of the point of intersection of tangents at $P$ and $Q$ from the ...
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1answer
215 views

What is the geometry behind $\frac{\tan 10^\circ}{\tan 20^\circ}=\frac{\tan 30^\circ}{\tan 50^\circ}$?

This identity is solvable by help of trigonometry identities , but i think there is an interesting and simple geometry interpretation behind this identity and i can't find this. Edit : I find ...
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33 views

Locus of circumcentre

Let $ABC$ be a triangle, and $P$ a variable point on its circumcircle. Suppose $AP$ meets $BC$ at $Q$. What is the locus of the circumcentre of $\triangle BPQ$? Experiments on GeoGebra show that the ...
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4answers
50 views

Equation of a circle tangent to two lines , given the radius . [closed]

What is the equation of the circle whose center is in the first quadrant and with the radius of $4$ units, given that it is tangent to the $x$-axis and to the line $4x-3y=0$?
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2answers
74 views

Proving $AE+AP=PD$ In a Certain Right Triangle

$\angle B$ in $ \triangle ABC$ is right. The incircle of $ \triangle ABC$ is tangent to the side $AB,BC,CA$ in $E,D,F$. The line $AD$ meets the incircle of $ \triangle ABC$ in $P(\neq D)$. If ...
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1answer
26 views

A similarity of $\Bbb Q^2$ without a fixed point

It is well known that any contraction of $\Bbb R^2$ has a fixed point. In particular, every similarity with the constant different from $1$ has a fixed point. The proof makes use of Banach fixed-point ...
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1answer
16 views

Hyperbola with its directrix

The equation $9x^2 - 16y^2 -18x +32y-151=0$ represents a hyperbola . We have to find the equation of its directrix. I simplified the equation and got : $$(3x-1)^2 -(4y-1)^2 = 151$$ And found that ...
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1answer
72 views

A construction with ruler and rusty compass

In the book Geometry: Euclid and beyond, the exercise 2.20 says: Using a ruler and rusty compass, given a line $l$ and given a segment $AB$ more than one inch long, construct one of the points $C$ ...
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22 views

Probability function of Euclidean distance between 2 vectors and origin?

In two-dimensional ($\mathbb R^2$) real space, if I have two vectors $x \sim N(3,1)$ independent of another vector $y \sim N(0,1)$, then what's the probability using R that the distance from origin to ...
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20 views

Surfaces with self intersection in 3-space

Let $F$ be a sphere in the Euclidean 3-space $\mathbb{R}^3$ With self intersection. Let $C$ be a double point circle in $F$. Then the double circle $C$ must bound a 2-disk in the standard sphere ...
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1answer
15 views

Prove that the bisectors of the 4 interior angles of a quadrilateral form a cyclic quadrilateral.

I can't seem to draw a good diagram for this question. I tried to draw a quadrilateral and draw the angle bisectors, but they intersected to form a very small quadrilateral. Then I tried to draw a ...
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1answer
16 views

dimension of space of origin-symmetric ellipsoids

I wonder how I can compute the dimension of the space of all origin-symmetric ellipsoids? What is the best approach?
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1answer
17 views

Do Euclidean geometry preservers parallelism of lines and area ratios?

Do the Euclidean geometry preserves the properties parallelism of lines and area ratios for any possible transformation? I know that the Affine geometry do and I think that Euclidean geometry also do ...
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2answers
41 views

Coordinates of circumcentre of an isosceles triangle in 3D

I have an isosceles triangle in 3D and I need to find the coordinates of the circumcentre of this triangle. I know the coordinates of the three vertices. One method I thought of is to solve equation ...
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2answers
46 views

Proving that four lines (which are perpendicular bisectors of chords) meet a point

In the diagram above, each of the four lines is a perpendicular bisector of one of the circles' chord. There are two pairs of circles which touch each other, and of course, as shown in the diagram, ...
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34 views

Are 2 quadrilaterals similar if they are both inscribed and have congruent angles and have perp diagonals

This is problem 365 from Kiselev's Planimetry book. I have to show that two inscribed quadrilaterals with perpendicular diagonals are similar iff they have respectively congruent angles. Here is my ...
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2answers
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Constructing a line that passes through $P$

I have recently read a book by Heisuke Hironaka. However, the book is not available on English. The book was basically a biography on his life. Heisuke Hironaka says that his high school teacher had ...
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4answers
310 views

Can you find the treasure??

My big bro gave this problem one week ago. I could not still solve it.Please HELP. STORY A man was just looking for items in his store room. Suddenly he found a map , which showed then it stated ...
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1answer
45 views

How likely is it that a random plane through the origin will intersect positive space?

In an n-dimensional hyperspace, how likely is it that a randomly chosen plane passing through the origin will intersect "all-positive co-ordinate space"? (By "all-positive co-ordinate space" I mean ...
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2answers
56 views

Can you construct a rectangle with a given side, equal to a square?

In Euclid's Elements, Book 2, Proposition 14, We are shown how to construct a square from a given rectilinear figure. This allows us to square a rectangle. Is it possible to do the inverse, creating ...
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1answer
71 views

An object is placed in front of a plane mirror of length $L$ …

I am stuck on the following problem : An object is placed in front of a plane mirror of length $L$ at a distance $d$ of its bisector line .An observer is at a perpendicular distance of $2d$ from ...
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1answer
21 views

How to prove that a regular hexagon covers the euclidian plane without overlapping and gaps?

My question is: How to prove that a regular hexagon covers the euclidian plane without overlapping and gaps? I think it would be the same as proofing the case that an equilateral triangle is ...
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16 views

axiomatic Euclidean geometry and its relation to the geometry of special relativity

It has been shown that the Euclidean plane defined by Hilbert's axioms is isomorphic to the 2D Euclidean vector space. Spacetime in special relativity can't be modeled by an Euclidean vector space, so ...