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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1answer
28 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 ...
4
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2answers
90 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
44 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
31 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
45 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
29 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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2answers
300 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 the help of trigonometry identities, but I guess there is an interesting and simple geometry interpretation behind this identity and I can't find it. I found it when ...
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0answers
38 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
97 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
88 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
30 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 ...
0
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1answer
19 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
100 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
24 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
22 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
33 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
18 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
21 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
50 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
51 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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0answers
46 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
103 views

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
328 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
47 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
94 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
75 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
24 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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0answers
20 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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2answers
41 views

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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1answer
26 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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2answers
32 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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0answers
12 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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0answers
20 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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1answer
20 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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1answer
65 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
89 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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0answers
23 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 ...
5
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1answer
67 views

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
32 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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1answer
115 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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1answer
18 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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2answers
61 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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1answer
78 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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2answers
75 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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0answers
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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1answer
28 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 ...
2
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1answer
33 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
35 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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2answers
138 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 ...