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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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 ...
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parallel postulate of Euclidean geometry and curvature

In elementary geometry, we have two standard examples which violate the (strong) parallel postulate of Euclidean geometry: in hyperbolic geometry, we have more than one parallel through a point which ...
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20 views

Find length of arc by projecting its points vertically?

Even though, algebraically, it is obvious that projecting the points of an arc vertically to the x-axis to find its length doesn't work, which postulate states that you cannot do that? Here's an ...
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26 views

Simple Euclidean Norm Inequality

I feel rather silly for having to ask this question in specific and am by no means looking for a flat out step by step answer. I understand the definition for the euclidean norm in an n-dimensional ...
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10 views

Diagonal of parallelepiped circumscribed around ellipsoid is constant

There are many rectangular parallelepipeds that can be circumscribed around a given ellipsoid in $\mathbb R^n$. Prove that the length of the main diagonal does not depend on the choice of such ...
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19 views

Unique tangent to conics

Given a conic section $C$ it is easy to prove analytically (or algebraically) that there is a unique tangent to $C$ in each point. Is there a simple synthetic proof of this fact? References are also ...
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17 views

Comparing areas of different parallelograms with same sides

Suppose I have parallelograms of same sides say 5 and 10 units with different left-bottom angle as $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, $\frac{\pi}{2}$. What is the comparison between ...
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58 views

Using Radical Axis to prove Concurrence

Let $BB',CC'$ be altitudes in $\triangle ABC$, and assume $AB\neq AC$. Let $M$ be the midpoint of $BC$, $H$ the orthocenter of $\triangle ABC$, and define $D$ as the intersection of lines $BC$ and ...
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5answers
81 views

Show that the angles satisfy $x+y=z$

How can I show that $x+y=z$ in the figure without using trigonometry? I have tried to solve it with analytic geometry, but it doesn't work out for me.
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20 views

Decompose cyclic sum of crossproducts into two cyclic sums?

Suppose you have $6$ points $a_i\in\mathbb{R}^3$ $i\in\{1,..,6\}$ such that all triangles with vertices $0, a_i, a_{i+1}$ for $i\in\{1,..,5\}$ do not degenerate (I dont know if this assumption is ...
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59 views
+50

Extend isometry on some cube vertices to the entire cube

Let $K\subset V=\{-1,1\}^n$ be a set of vertices of the $n$-dimensional hypercube $D=[-1,+1]^n$ and let $f:K\to V$ be an isometry with respect to the Euclidean metric inherited from $\mathbb R^n.$ We ...
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1answer
25 views

Finding a curve from its evolute

Consider the evolute given by $$ \gamma: I \subset \mathbb{R} \to \mathbb{E}^2: t \mapsto (\cos(t),\sin(t))$$ Now, how do I find all the curves $\alpha$ that have $\gamma$ as their involute? I tried ...
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83 views

Prove that a straight line is the shortest distance between two points?

Prove that a straight line is the shortest distance between two points in $E_3$. Use the following scheme; let $\alpha: [a,b]\to E_3$ be an arbitrary curve segment from $p = \alpha(a) , q = ...
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17 views

Formally showing that there exist exactly four isometries of $\mathbb{E}^2$ that map two intersecting lines

Given are two intersecting lines $l$ and $l'$ in $\mathbb{E}^2$. How does one show that there are exactly four isometries that map $l$ to $l'$ and have $l\cap l'$ as fixed point? Intuitively, I've ...
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59 views

Euclidean Geometry

$XYZ$ is a triangle in which $\angle X$ is obtuse. A point $P$ is taken inside the triangle and $XP$, $YP$, $ZP$ are produced to meet the sides $YZ$, $ZX$, $XY$ at the points $K$, $L$, $M$, ...
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56 views

Is there really no proof to corresponding angles being equal?

I've read in this question that the corresponding angles being equal theorem is just a postulate. However I find this unsatisfying, and I believe there should be a proof for it. However the only way ...
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60 views

Locus of intersection of two lines

If the tangent at any point P of a circle $x^2 + y^2 = a^2$ meets the tangent at a fixed point A $(a,0)$ in T and T is joined to B , the other end of the diameter through A . Then we have to prove ...
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19 views

Existence of solution of the system of inequalities

I want to find a simple way to determine, if the following system of inequalities has a non-trivial solution : $a_{1,1}x_1+a_{1,2}x_2+\ldots+a_{1,n}x_n \le 0$ ...
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20 views

A point $P(a,b)$ is equidistant from the y-axis and from the point $(4,0)$. Find a relationship between $a$ and $b$.

A point $P(a,b)$ is equidistant from the y-axis and from the point $(4,0)$. Find a relationship between $a$ and $b$. I know that the distance of $(a,b)$ from the point $(4,0)$ is $\sqrt ...
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1answer
25 views

What is the probability density function of pairwise distances of random points in a ball?

Suppose that one selects two random points x,y in a sphere of radius R. Is there a closed-form expression for the probability density P(d_x,y), i.e. the probability that x and y have Euclidean ...
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1answer
33 views

Using predicate logic to verify the theorems of Euclid's elements?

I wanted to make a "logical" march through the entirety of Euclid's elements by proving and verifying, step by step, each theorem using Hilbert's axioms as a basis. Of course, I would want to do this ...
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129 views

What is the minimum polynomial of $x = \sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6} = \cot (7.5^\circ)$?

Inspired by a previous question what let $x = \sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6} = \cot (7.5^\circ)$. What is the minimal polynomial of $x$ ? The theory of algebraic extensions says the degree is ...
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32 views

Projection of a point on a set

Let $M=\{x\in\mathbb{R}^{3}: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\le4, x_{1}^{2}-4x_{2}\le0\}$ and $y=(1,0,2)^{T}$. What is the projection of the vector $y$ on the set $M$?
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125 views

Derivative of intersection volume

Let $K$ be a convex body in $\mathbb{R}^n$ and set $f:\textrm{SL}(n)\rightarrow \mathbb{R}$ as $f(T)=\textrm{Vol}_n (TB\cap K)$ where $B$ is the Euclidean unit ball. How can we find extreme points of ...
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47 views

How can you compute the maximum volume of an envelope(used to enclose a letter)? [closed]

It's obvious that the volume of a envelope is 0 when flat and non-0 when you open it up. However, if you were to fill it with liquid, there must be some shape where it has a maximum volume. Is there a ...
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1answer
52 views

Center of Arc with Two Points, Radius, and Normal in 3D

I'm struggling to get the math to work out on this. I need to derive an alorithm for a program where I'm representing geometric entities. In this case, it's an arc. I would like to create the arc ...
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1answer
39 views

Locus of vertex of a rectangle

If from the vertex of a parabola $y^2 = 4ax$ a pair of chords be drawn at right angles to one another and with these chords as adjacent sides a rectangle be constructed , then we have to find the ...
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63 views

Proving result in inscribed triangles.

ABC is a triangle inscribed in a circle, and E is the mid-point of the arc subtended by BC different from the arc A on which A lies. If through E a diameter ED is drawn, show that $$\angle ...
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1answer
23 views

Normal to a parabola

If we have to find normal to the $x^2 = 4by$ I tried Let a point be $(-2bm,bm^2)$ Then equation of tangents is $-2bmx = 2b(y + bm^2)$ hence slope is $-2m$ So the slope of normal is ...
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49 views

The point of intersection of two perpendicular tangent lines to a parabola

If two perpendicular straight lines through the focus of the parabola $y^2 = 4ax$ meet its directrix in $T $ and $T'$ respectively. Show that the tangents to the parabola parallel to the ...
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31 views

Maximising the ratio area/perimeter [duplicate]

I read (in a book of physics using a not always rigourous language) that the circle maximises the ratio $$\frac{S}{L}$$where $S$ is the area and $L$ the perimeter among plane figures. I find the ...
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Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$

Compute $d(x^{100},P_{\le 98})$ where $P$ is subspace of polynomials with degree $\le 98$, looking at $C_{(2)}[-1,1]$, with $L_2$ norm. I tried to look at a general polynomial $\sum_{i=0}^{98} ...
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27 views

From a point to the Vertex

I was aked to solve the following problem: Guiven three lenghts and a triangle ABC, from every vertex whe draw one of the three lenghts, find the conditions such that the three lenghts meet in one ...
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2answers
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Is my definition of a triangle center function “equivalent” to the usual definition?

There's a definition of triangle center function already in existence, but I don't really understand it. Anyway, here's my attempt at defining this concept using ideas I'm more comfortable with. ...
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37 views

Prove the following statement

Prove that the point $P=(x',y')$ will be inside the acute or obtuse angle made by $a_1x+b_1y+c_1= 0$ and $a_2x+b_2y+c_2= 0$, if $$(a_1x'+b_1y'+c_1)(a_2x'+b_2y'+c_2)(a_1a_2+b_1b_2)\quad <\text{ or ...
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287 views

Is the point of a shape with the greatest average ray length also the “centroid”?

By a shape I mean one with a enclosed area. One example can be with $x^2+y^2+\sin{4x}+\sin{4y}=4$. I am dealing with implicit relations on a 2-d plane. By the "ray length" I mean the length of the ...
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Inscribe square in circle in just seven compass-and-straightedge steps

Problem Here is one of the challenges posed on Euclidea, a mobile app for Euclidean constructions: Given a $\circ O$ centered on point $O$ with a point $A$ on it, inscribe $\square{ABCD}$ within the ...
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39 views

proof for the square root construction

I know that you can construct a square root of a given segment through a certain construction, and I know that construction. What is the proof that this construction works to construct a square root? ...
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39 views

Building Euclidean space

What's the minimum amount of extra "structure" do we need to add to the general concept of an affine space to get Euclidean space? That includes the concepts of angle and distance, in which we can ...
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26 views

How does a group of transformations lead to a geometry?

I am reading Vinberg's algebra text, and on page 144 he says "Of course, not every transformation group leads to a geometry which is interesting and also important for some applications. All such ...
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What is the simplest way to find $\frac{n}{7}th$ of a line with mathematical proof?

I'd like to know if it is possible to find out the simplest way to get $\frac{1}{7}$, $\frac{2}{7}$, $\frac{3}{7}$, $\frac{4}{7}$, $\frac{5}{7}$ and $\frac{6}{7}$ of a line in 2-dimensional geometry. ...
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Chord of a parabola

If the end points $P(t_1)$ and $Q (t_2)$ of a chord of a parabola $y^2 = 4ax$ satisfy the relation $t_1\cdot t_2 = k$ (constant) then we have to find the fixed point through which chord will always ...
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1answer
36 views

Parabola and straight line [duplicate]

If $m$ varies then find the range of $c$ for which the line $y=mx + c$ touches the parabola $y^2 = 8(x+2)$ . I tried Put the value $y = mx + c$ in the parabola equation and then done $\Delta = 0$ or ...
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1answer
57 views

Finding and proving similar triangles

ABC is a triangle with AB shorter than side AC. The angle bisector of ∠A intersect BC at D. Given that point E is on the median that's drawn from A, so that BE⊥AD, how do I show that DE||AB? I tried ...
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1answer
28 views

Coordinate-free expression of a rotation

I'm interested in coordinate free (non-matrix based) approaches to geometry. What I'd like to do is to show that every Galilean transformation can be written uniquely as the composition of a ...
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In Euclid' s Elements, why is $\frac{\sqrt{a^ 2 - b^2}}{a}$ important in the definition of “apotome”?

Although I had originally found them in Book X of Euclid's Elements, there is a summary of the definition of apotome on the web. After sifting through the definitions there are 4 conditions Euclid ...
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Dual set of the unit ball is part of the unit ball.

Define the unit ball centered at the origin as $B=\{x\in\mathbb{R}^d\mid \|x\|\leq 1\}$. Define the dual set of set $X$ as $X^*=\{y\in\mathbb{R}^d\mid\langle x,y \rangle\leq 1\ \forall x\in X\}$. ...
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46 views

Stick passing through glass leaves a possibly parabolic hole

I am trying to understand what happens in this gif video: Source: http://9gag.com/gag/aAVp4V9/is-this-even-possible It is quite interesting because at a first look, it was very counter-intuitive. ...