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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The rectangle-partition number and the number of horizontral edges

The rectangle-partition-number of a rectilinear polygon $P$ is the smallest number of pairwise-disjoint axis-parallel rectangles required to cover $P$. Some examples: (in the last example, $P$ is ...
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23 views

Proving that union of epsilon nets is an epsilon net

While reading a paper, I came across these definitions and claims: Definition: Given $p \in \mathbb R^d$, and $H$, a set of hyperplanes, let $\text{Violate}_p(H) = \{h \in H: h \text{ is strictly ...
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38 views

Synthetic geometry , angles.I need some ideas

Let $ABC$ be a triangle such that $m(\measuredangle ACB)>30$ and $M$ in the interior of the triangle with $m(\measuredangle BMA)=120, m(\measuredangle BCM)=30$. Let$\{ D\} = AM\cap BC$ and $P \in ...
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15 views

Property of all k-dimesional shapes

A close friend showed me an intuitive pattern that given a k-dimensional convex polytope P, if one doubles every sidelength to generate a $P'$ it is possible to fit $2^k$ copies of $P$ within the ...
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21 views

Concurrence of lines through triangle midpoints [duplicate]

Suppose that points $A, B, C$ form a triangle and that $A'$ $\in$ $l_{BC}$ and $B'$ $\in$ $l_{AC}$ and $C'$ $\in$ $l_{AB}$ are such that the lines $l_{AA'}$, $l_{BB'}$, $l_{CC'}$ are concurrent at G. ...
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39 views

Concurrence through a triangle

Suppose that points A, B, C form a triangle and that A' $\in$ $l_{BC}$ and B' $\in$ $l_{AC}$ and C' $\in$ $l_{AB}$ are such that the lines $l_{AA'}$, $l_{BB'}$, $l_{CC'}$ are concurrent at G. ...
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1answer
27 views

Two new conics associated with two triangles inscribed a conic

1-Let $ABC$ and $A'B'C'$ be two triangle inscribed a conic. Let $P$ be a point on the conic. $PA$ meets $B'C'$ at $A_1$, define $B_1, C_1$ cyclically. $PA'$ meets $BC$ at $A'_1$ define $B'_1, C'_1$ ...
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1answer
50 views

$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could then the $\pi$ also be an integer?

$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could there be a geometry where $\pi$ is a rational number or an integer?
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2answers
60 views

Prove that $u\cdot v = 1/4||u+v||^2 - 1/4||u-v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$

I need some help figuring out how to work through this problem. Prove that $ u \cdot v = 1/4 ||u + v||^2 - 1/4||u - v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$. Sorry, forgot to ...
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36 views

Common Intersection Point of ellipsoids

Suppose we have two ellipses in $2$-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have $4$ points of intersection. Is it known ...
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21 views

locus of a variable straight line [closed]

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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1answer
19 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
38 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
26 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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16 views

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
30 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 ...
3
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1answer
50 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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0answers
20 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
43 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
28 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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23 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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28 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
122 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
46 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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0answers
32 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
86 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 ...
11
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1answer
232 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 it. I found it when I ...
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0answers
34 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
51 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
77 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
27 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 ...
5
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1answer
73 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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0answers
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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0answers
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 ...