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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Assuming that the sum of the angles of any triangle is 180, how can I deduce Euclid's 5th postulate?

I already did the reverse, namely, if we assume Euclid's 5th postulate, then the sum of the angles of any triangle is 180 degrees. Now I need to show the converse, but I don't really know how to ...
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1answer
65 views

Geometric inequality $180^{\circ}\left(1-\frac1n\right)\le \angle{AMB}$

Point $M$ is located inside a regular $n$-gon. Prove that there exist different vertices $A$ and $B$ that $$180^{\circ}\left(1-\frac1n\right)\le \angle{AMB}\le 180^{\circ}$$ My work so far: Let $...
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1answer
61 views

Simple Golden Ratio Construction with Three Lines, and Interesting Implied Circle?

Consider three equal lines (as illustrated below). A red, green, and orange line of equal length all rest upon the same horizontal line. The red line is stood upon its end in a manner perpendicular ...
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4answers
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Golden Ratio Conjecture in three simple Geogebra shapes--circle, triangle, and square.

A circle, equilateral triangle, and square of equal heights are all placed on the same horizontal line as shown below. The circle is tangent to the triangle which is centered upon the left edge of ...
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1answer
84 views

Two Equilateral Triangles and the Golden Ratio: Simple Geomtric Proof

Two equilateral triangles rest upon the same horizontal line, as shown in the linked figure below. The red triangle is tilted so that one corner touches a side of the black triangle at the midpoint ...
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1answer
25 views

Is this definition of a Euclidean frame well-defined?

Going through my lecture notes on geometry I find a definition of a Euclidean frame which doesn't seem to have been formed correctly (most likely written down wrong). So I've taken it upon myself to ...
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1answer
32 views

Angles in Hilbert's axioms for geometry

In Hilbert's axioms for geometry, the following elements are presented as undefined (meaning "to be defined in a specific model"): point, line, incidence, betweenness, congruence. In deed, when ...
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1answer
19 views

How to fit a convex quadrilateral inside another short of cutting them out and playing with them?

I have two convex quadrilaterals (ABCD and WXYZ). Their sides and their interior angles are known. I also know that WXYZ fits perfectly inside ABCD with each corner point touching a different side. ...
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1answer
120 views

Water filling problem in Blocks - Algebra Question

Consider a rectangular plot comprising $n\times m$ square cells on which $nm$ cement blocks of various heights are stored, one block per cell. The base of each block covers one cell completely, and ...
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2answers
51 views

Power of a point proof

I found the question on page 13 of this link. Let $P$ be a point inside a circle such that there exist three chords through $P$ of equal length. Prove that $P$ is the center of the circle. I ...
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Can a figure inside a circle be seen at right angle from any point on the circle?

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex ...
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1answer
25 views

Isometries in $\mathbb{E}^2$

We define $\mathbb{E}^2 = (\mathbb{R}^2, d_E)$, where $d_E$ is the usual Euclidean metric on $\mathbb{R}^2$. We say that a function $f:X \rightarrow Y$, where $X, Y$ are metric spaces, is an isometry ...
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42 views

An elementary problem in Euclidean geometry

Let $ABC$ be an acute triangle ($AB < AC$) which is circumscribed by a circle with center $O$. $BE$ and $CF$ are two altitudes and $H$ is the orthocenter of the triangle. Let $M$ be the ...
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1answer
62 views

Draw A Triangle From 3 Excenters and Ex-radii

My teacher gave me this problem and told me to think- " Is it possible to draw a triangle, given the three ex-centers and length of the ex-radii?" I don't know if it's possible or not. So, my ...
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1answer
59 views

How to show $\partial A = \varnothing \Rightarrow A=R^n$ [closed]

Let $A\subset R^n$ and dim$A=n$, $\partial A$ is the relative boundary of $A$. If $\partial A=\varnothing$ how to show $A$ is $R^n$ ? Picture below is from XX page of Schneider R.-Convex Bodies_ ...
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Prove $∠ADM = ∠ACB$ of triangle $ABC$ [closed]

Suppose that $ABC$ is a triangle. Let $D$ be its circumcenter and let $M$ be the midpoint of $\vec {AB}$. Show that $∠ADM = ∠ACB$.
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1answer
25 views

Every orthogonal matrix represents a rotation around an axis

Is it true that every element of the group $O(n)$ represents a rotation around some axis? I'd like this to be true in order to decompose any matrix $R \in O(n)$ as a block matrix in $O(n-1)$ and a 1 ...
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15 views

How to determine the angle from a point and the plane tangent points in a sphere

I have an UAV modeled in three dimensions with let's say position coordinates $p_{uav} = (x_1,y_1,z_1)$ that is moving in a direction $d = (d_x,d_y,d_z)$ and a moving obstacle modeled as a sphere with ...
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1answer
73 views

How to prove a regular pentagon is formed by knotting a rectangular strip of paper?

I found this interesting problem from a friend (From Arthur Engel's-Problem Solving Strategies book). The method to begin the problem is as follows- Step 1.Take a rectangular strip of paper ...
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1answer
23 views

There are finitely many $n$-dimensional wallpaper groups

There are $7$ Frieze groups, $17$ wallpaper groups and $230$ space groups, which are the 1,2,3-dimensional cases of isometries on $\Bbb R^n$ (I think? Frieze groups seem to require $[0,1]\times\Bbb R$ ...
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Prove a harmonic range from a familiar picture

During solving some simple problem (10th grade), I found this interesting problem, which I got no clue to solve it clean and properly. Hope someone can give me some hint to solve it. Thanks. Given ...
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Are given maps isometric?

I'm trying to determine if certain maps are isometric in $\mathbb R^2$. The two maps I have to analyze are f such that: $|f(X)| = |X|$ $f(X)*f(Y) = X * Y$ where $*$ is the dot product (inner ...
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28 views

Convert a 3d vector into a rotation matrix?

Is it possible to compute a Rotation matrix given a 3d vector given in the Euclidean space? and if not what would it need? An illustration of my situation. Illustration of my problem I have a ...
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Right triangle inscribed in a square. Find the square area?

I hope it's valid to ask for (a more neat solution) of a problem on this network, despite the fact that I don't have a strict definition of the word "neat". Here is the square and the right triangle ...
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How do I convert an x,y,z to an Q configuration?

I am trying to implement a tracking application for a robot arm, which purpose is relocate itself based on the position of an object seen from the tool point. The robot arm itself is a UR5, and at ...
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1answer
45 views

Prove that $DQ \times DB = DP \times DC + DR \times DA$.

Let $ABCD$ be a parallelogram, with $P$, $Q$, and $R$ the points on which a given circle passes through $D$ and cuts through the segments $CD$, $BD$ and $AD$ respectively: How do you prove that $DQ ...
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1answer
57 views

Ellipse's farthest point to another point

I am trying to find the farthest and closest points of a ellipse without using any brute force type of coding. The processing power is limited so it should be as pinpoint as possible. I have tried a <...
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1answer
33 views

Moving Line Segment Problem part 2

This question is related to a question I asked a while ago here on math.stackexchange: Moving Line Segment Problem The rules for how the line segment can be moved are the same: The endpoints must ...
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0answers
15 views

Isogonal Conjugate of point outside of triangle

I was wondering about reflections of lines over the external bisectors instead of external bisectors in a triangle. Here is a problem that brought it up: Let $P$ be a given point inside quadrilateral ...
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0answers
32 views

Volume of a cone [duplicate]

I have a simple question about the formula for the volume of the cone. Let $C$ a cone, which base has radius $r$ and height equal to $h$. So its volume can be compute by the formula: $$\text{Vol}(C)=\...
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1answer
32 views

Prove perpendicular bisectors of non-parallel lines intersect

Suppose that $A$, $B$ and $C$ are points and that $AB$ and $BC$ are not parallel. Show that the perpendicular bisector of $AB$, $l$, and the perpendicular bisector of $BC$, $l'$, are not parallel and ...
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Prove Concurrency using Radical Axis of Circumcircles

Let the incircle of $\triangle ABC$ touch sides $BC,CA,AB$ at $D,E,F$, respectively. Let $\omega,\omega_1,\omega_2,\omega_3$ be the circumcircles of $\triangle ABCm,\triangle AEF,\triangle BDF,\...
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340 views

Optimal path around an invisible wall [duplicate]

The Problem On an infinite plane there are two points, $A$ and $B$, a unit distance apart. There is a $50\%$ probability that there is an invisible wall somewhere between the two points. The ...
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1answer
16 views

Characterizing isotropic measures

A Borel measure $\mu$ on $S^{n-1}$ is called isotropic if $$\int_{S^{n-1}} \langle \theta, x \rangle^2 d\mu(x)=\frac{\mu\left({S^{n-1}}\right)}{n}$$ for all $\theta\in S^{n-1}$. This means that in ...
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3answers
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Axioms of Geometry?

I have taken the generic low level undergraduate classes, such as Calculus 1-3, Differential Equations, and Linear algebra. Since I never learned Geometry past a basic high school level, I thought it ...
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27 views

Prove that DE || BC

Let M be the midpoint of side BC in triangle ABC. The angle bisector of BMA intersects AB in D, while the angle bisector of CMA intersects AC in E. How can i prove that DE||BC? I drew out the ...
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1answer
39 views

Finding the set of points on the sphere with an equal product of distances

Given two points $x_1$ and $x_2$ on the sphere, one can find another set of points on the sphere $\{y_1, y_2\}$ such that the product of Euclidean distances to the given points $x_i$ is the same for ...
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How many Pascal hexagons can I construct with 6 different points on a circle?

I have a basic knowledge about combinatorics and I am in a euclidean geometry class. My question is : How many Pascal hexagons can I construct with 6 different points on a circumference? It could ...
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1answer
38 views

Name of the geometric figure of points ${\bf x} \in \Bbb R^n$ with $1$-norm $||{\bf x}||_1 = 1$

Is there a name for the figure $$\{{\bf x} \in \Bbb R^n : ||{\bf x}||_1 = 1\} \subset \Bbb R^n ?$$ Things like this seem to usually have names, for instance, the $n$-cube or $n$-ball. In $2$ ...
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what is the value of angle A

The triangle ABC is random. The line $AD$ is twice big as the line $DC$ ($AD=2*DC$). We know only the two angles that are shown in the picture. What's the value of angle $A$?
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1answer
31 views

Median BM of triangle ABC two results

Given Calculate the measure of the median $\overline{BM}$ of ABC triangle, given A (-6.1); B (-5,7) and C (2,5) I get this result: $Xm = \frac{Xc - Xa}{2} + Xa$ $Xm = \frac{2-(-6)}{2} + (-6) = ...
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11answers
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In a right triangle, can $a+b=c?$

I understand that due to the Pythagorean Theorem, $a^2+b^2=c^2$, given that $a$ and $b$ are legs of a right triangle and $c$ is the hypotenuse of the same right triangle. However, most of the time, $a+...
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1answer
37 views

Is $f(x) = x^2$ a scalar function?

Take a simple parabola. It is a function that has a one-dimensional co-domain $$f(x) = x^2 $$ It is mapping the set of values in its domain, to one-dimensional values in its co-domain, and it ...
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Why this function describes a euclidean ball?

In Stephen Boyd's convex optimization book at page 97, one can read : $$ a,b \in R^n $$ $$ (1-\alpha^2)x^Tx-2(a-\alpha^2b)^Tx+a^Ta-\alpha^2b^Tb \leq 0 $$ is convex (in fact a euclidian ball) if $ \...
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1answer
24 views

Find distance of overlapping squares

How to find distance center to center from square $1$ to square $3$, if we need overlap area is $15.46 mm^2$. if we know each side of the square is $6.9 mm$. Firstly I find the distance is $9.3 mm$ ...
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Bounds on the size of Voronoi cells

I am working on an algorithm for which bounds on the size of voronoi cells will come in handy. Suppose that the domain $D$ is partitioned according to the Voronoi cells $D_1,\dots,D_n$ with Voronoi ...
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1answer
18 views

Perturbation of tangent ball

As picture below, $A$ and $B$ are two balls, $\partial A\bigcap \partial B=\{k\}$, and $B$ contains $A$. How to show that $$ \forall h\in \partial B,\exists ~\varepsilon > 0 ~st~ A\subset B+\...
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2answers
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Is there is any other method to produce a third set of collinear points rathar than the Pappus's hexagon method?

Pappus's hexagon theorem: Given one set of collinear points $A,B,C$, and another set of collinear points $a,b,c$, then the intersection points $X,Y,Z$ of line pairs $Ab$ and $aB$, $Ac$ and $aC,Bc$ and ...
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Prove that the intersection point of lines $AK$ and $CL$ lies on the line $BO$

$AA', BB'$ and $CC'$ heights of an acute triangle $ABC$. The circle with center $B$ and radius $BB'$ intersects the line $A'C'$ in the points $K$ and $L$. Prove that the intersection point of lines $...
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nearest neighbour graph, euclidean distance and graph Laplacian

It could be a very simple problem and someone will prove an answer in a second. So, I have unlabelled data points $\left((-2,0),(-1,-1),(0,0),(1,1),(2,0)\right)$ and would like to to the following: ...