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.

learn more… | top users | synonyms (1)

0
votes
1answer
13 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 ...
2
votes
3answers
56 views

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 ...
0
votes
2answers
24 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 ...
1
vote
1answer
31 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 ...
0
votes
0answers
23 views

Finding the number of lattice points enclosed by a right triangle with non-integer coordinates.

I would like to find the number of lattice points enclosed by a triangle with coordinates $(0,0)$, $(0,a)$, and $(b,0)$. I Assumed that I would merely need to truncate $a$ and $b$, and apply and use ...
0
votes
0answers
25 views

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 ...
1
vote
1answer
36 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 $2$ ...
2
votes
5answers
68 views

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 $A$?
1
vote
1answer
28 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) = ...
19
votes
11answers
5k views

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, ...
0
votes
1answer
36 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 ...
4
votes
1answer
65 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 $ ...
0
votes
1answer
17 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 $9.3 mm$ ...
2
votes
0answers
15 views

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 ...
1
vote
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 ...
0
votes
2answers
41 views

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 ...
2
votes
0answers
51 views

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 ...
1
vote
0answers
32 views

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: ...
2
votes
1answer
18 views

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 ...
0
votes
0answers
12 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 ...
0
votes
0answers
12 views

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, ...
0
votes
0answers
14 views

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 ...
2
votes
1answer
31 views

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 ...
1
vote
1answer
59 views

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 ...
0
votes
0answers
9 views

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 ...
1
vote
3answers
52 views

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. ...
1
vote
2answers
75 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 ...
1
vote
2answers
33 views

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 ...
7
votes
0answers
155 views

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. ...
-1
votes
1answer
27 views

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 ...
1
vote
1answer
12 views

Prove or disprove that any three members of a family of parallelograms intersect

Given a family of parallelograms such that the corresponding edges of all members are parallel and any two members of this family intersect. Can we conclude that any three members of this family ...
1
vote
1answer
31 views

Helix along vector in 3D space

Let's say I have a random vector, for example <1, 3, 5>. What would the function be for a helix that spirals around/along this vector with a given radius?
0
votes
1answer
18 views

Finding geodesics of a surface

I'm not able to understand how the answer given above has been obtained. How did they deduce the sigma is isometric to the plane? Also, if a surface is isometric to another surface then does that ...
1
vote
4answers
52 views

Coordinate Geometry: Are there enough information to find out the coordinates?

Question: Given the circle $x^2+y^2=25$ is inscribed in triangle $\triangle ABC$, where vertex $B$ lies on the first quadrant. Slope of $AB$ is $\sqrt 3$ and has a positive y-coordinate, and ...
0
votes
1answer
19 views

Reflections in Angle bisector

In a triangle $ABC$, take the tangent to the circumcircle of $ABC$ at $A$. Reflect this line through the angle bisector at $A$. prove that this reflected line is parallel to $BC$. I'm looking for a ...
13
votes
2answers
129 views

$a^x+b^x=c^x$ in geometry

The Pythagorean theorem. Let $A$, $C$, $B$ be three points on a line in this order, and let $D$ be another point, such that $\angle ADC =\angle CDB = 60^\circ$. Let $a=AD$, $b=BD$, $c=CD$. Then, ...
1
vote
1answer
31 views

Locus of points $P$ on the plane such that $\overline{AP}=\lambda \cdot \overline{BP}$

Given two points on the plane $A$ and $B$ and given $\lambda \in (0,+\infty)$ consider the the locus of all the points $P$ such that $\overline{AP}=\lambda \cdot \overline{BP}$. If you study it with ...
2
votes
2answers
93 views

A chain of six circles associated with a cyclic hexagon

I found the problem some months ago. But I never have been a proof. So I am looking for a proof. The problem as following: Let $ABCDEF$ be a cyclic hexagon. Let $(C_{AD})$, $(C_{BE})$, $(C_{CF})$ ...
0
votes
1answer
19 views

Existence and uniqueness of a point with horizontal tangent in a convex curve

I had a look on the proof by E. Schmidt of the Schur's Theorem about arcs of convex curves. It states the following: Let $C$ and $C'$ be two arcs of the same length with the endpoints $a,b,a',b'$ ...
0
votes
0answers
20 views

Projective Geometry - Pole/Polar

A circle is inscribed in quadrilateral $ABCD$ so that it touches sides $AB, BC, CD, DA$ at $E, F, G, H$ respectively. (a) Show that lines $AC, EF, GH$ are concurrent. In fact, they concur at ...
1
vote
1answer
52 views

Is this a well defined problem in terms of Euclidean Geometry?

I am trying to construct an example of a geometric problem, stated in terms of Euclidean Geometry, that is not Machine Provable (or in an equivalent definition Automatically Provable)-i.e no computer ...
1
vote
1answer
27 views

Euclidean norm on integer lattice

Does the Euclidean $ L^2 $ norm (and distance) make any sense on an integer lattice in $ \mathbb{R}^n $? And what is the preferable way of calculating a type of norm in such spaces?
-1
votes
1answer
64 views

find $\angle{BDE}$

All information required is shown at the picture the only thing I could find is this equation: $\angle{BDE}$ + $\angle{CED}$ = 80 . I don't know what to do...
1
vote
0answers
49 views

Estimating the distance between two coordinates but without using Euclidean distance

Bill opens up "Café Finder" on his phone, and it tells him that it will take him 10 minutes to get to his nearest Starbucks to grab a triple-shot frapa-crapa-flat-white, so he decides to walk. 20 ...
1
vote
1answer
29 views

A straightedge and compass construction: $\left(\widehat{A},r,b-c\right)$

I am looking for an elegant solution of the following problem: Construct $ABC$ with straightedge and compass, given $\widehat{A},r,b-c$. By taking the lines $AB,AC$ as a skew reference system, ...
0
votes
1answer
18 views

Vector of triangle height constructed over two vectors

Given vectors: $\overrightarrow{a}=\overrightarrow{p}+2\overrightarrow{q},\overrightarrow{b}=3\overrightarrow{p}-\overrightarrow{q}$ where ...
0
votes
1answer
36 views

Suppose that $ABCD$ is a trapezoid with $AB$ parallel to $CD$.

Suppose that $ABCD$ is a trapezoid with $AB$ parallel to $CD$. Let $P$ be the point where the diagonals $AC$ and $BD$ intersect. Show that the triangles $CDP$ and $ABP$ are similar. Use this to prove ...
1
vote
1answer
24 views

How do I make an inner rounded rectangle and an outer rounded rectangle be parallel around the corners?

The outer radius does not follow the inner radius. I am currently using x = width/4 + radius + outset, y = height/4 + radius + outset. I think the outset needs to be some ratio of the ...
0
votes
1answer
28 views

Suppose that $a, b,$ and $c$ are distinct points in $\mathbb{C}$.

Suppose that $a, b,$ and $c$ are distinct points in $\mathbb{C}$. Let $l$ be the line which bisects the angle $\measuredangle bac$. For a point $z$ on $l$, let $p$ be the point on the line through $a$ ...
1
vote
1answer
29 views

Suppose that $a, b$, and $c$ are distinct points in $C$. Suppose that $l$ is the line which bisects $∠bac$ and $m$ is the line which bisects $∠acb$

Suppose that $a, b$, and $c$ are distinct points in $C$. Suppose that $l$ is the line which bisects $∠bac$ and $m$ is the line which bisects $∠acb$. Now let $z$ be the point $l \cap m$. Let $n$ be the ...