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 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
58 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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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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71 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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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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29 views

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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18 views

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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27 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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26 views

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
56 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
32 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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14 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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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
30 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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22 views

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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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
15 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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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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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
37 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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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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68 views

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
23 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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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: ...
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1answer
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All triangles that have the same orthocenter and circumcircle have the same nine-point circle

True or false? Prove it. I guess it would help to figure out whether 2 triangles can have the same circumcenter or orthocenter and not be congruent. I have no clue how to figure this out. If they ...
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14 views

A new family circle associated with the Tucker hexagon and the Symmedian point

I am looking for the problem following: Let ABC be a triangle, let $A_1B_1C_1$ be a cevian triangle of the symmedian point. Let $B_aC_aC_bA_bA_cB_c$ be a Tucler hexagon of $ABC$. Such that $A_bA_c ...
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A generalization of the first Droz-Frany circle

I am looking for a proof of the following problem: Let $ABC$ be a triangle with circumcenter $O$, and the medial triangle $M_aM_bM_c$. Let $O_a, O_b, O_c$ be three points on three lines $OA, OB, ...
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properties of a Varignon parallelogram from a skew quadrilateral,

I was editing https://en.wikipedia.org/wiki/Varignon's_theorem and that made me wonder. At the moment https://en.wikipedia.org/w/index.php?title=Varignon%27s_theorem&oldid=713877982 the ...
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What triangles can be cut into three triangles with equal radii of the circumscribed circles around these triangles?

What triangles can be cut into three triangles with equal radii of the circumscribed circles around these triangles? My work so far: Case 1) let $ABC -$ an acute-angled triangle. Then radii of the ...
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1answer
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Proving $\frac { { { A }_{ 1 }G } }{ G{ { B }_{ 1 } } } +\frac { { A }_{ 2 }G }{ G{ B }_{ 2 } } +…+\frac { { A }_{ n }G }{ G{ B }_{ n } } =n$

Let ${ A }_{ 1 }{ { A }_{ 2 }{ A }_{ 3 }...{ A }_{ n } }$ be a $n$-gon with centroid $G$ inscribed in a circle. The lines $\\ \\ { A }_{ 1 }G,{ A }_{ 2 }G,...{ A }_{ n }G$ intersect the circle again ...
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The smallest bounding sphere of a prolate spheroid domain

Let $\Omega\subset \mathbb{R}^3$ be a prolate spheroid domain. Denote by $d$ its interfocal distance and by $b$ the surface of the region occupied by $\Omega$. The question is how to prove that the ...
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Cover a polygon with polygons

Besides right angled triangles, is there any polygon I could use to cover any given (regular or not) polygon? It's clear that given a triangle, square, hexagon or rectangle you would other options. ...
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80 views

Geometry - Tangent circles

Let chords AC and BD of a circle ω intersect at P. A smaller circle ω1 is tangent to ω at T and to segments AP and DP at E and F respectively. (a) Prove that ray T E bisects arc ABC of ω. (b) Let I ...
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Construct orthogonal projection for plane (matrix form)

I am currently trying to get up-to-date with my university level math (i kinda slacked off a little), using some homework that our professor provied for us. Now, one task is like this (original ...
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Another chain of six circles

I found a conjecture: A chain of six circles associated with six points on a circle (in Mobius plane). This is a generalization of the last my previous question in Three chains of six circles. (...
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Find the angle between chords

I assume this is a simple problem, but I can't find the answer. I must don't know a theorem and went to a wrong direction. Problem: AC and BD are chords in a circle intersecting at E. If the measure ...