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)

1
vote
1answer
21 views

Specific triangle, symmetral of angle proof

In triangle $ABC$, angle $\gamma = 120$. Prove that $|\overline{CC'}|=\tfrac{ab}{a+b}$, where $\overline{CC'}$ is symmetral of angle $\gamma$ inside triangle. Look at image. I can't use areas, ...
2
votes
1answer
19 views

How to represent two coordinate system transformations as one

I'm working on a system of relative euclidean coordinate systems. I'd like to define every coordinate system relative to a global coordinate system, which I'll refer to as [0]. Then, for example, ...
1
vote
2answers
282 views

Bounding box enclosing circles, that complies with ratio constraints

Given a circle centered at $A$, with radius $R_a$ and another radius $R_b$, I need to find a center for circle $B$ such that both circles are tangential, and the bounding box including both circles ...
10
votes
1answer
334 views

The number of the circles which are tangent to two circles and to a line

Suppose that we have two distinct circles and a line on a plane and that the distance between the centers of the circles is bigger than the sum of their radiuses. Also, suppose that the two circles ...
1
vote
0answers
14 views

Dandelin spheres and the asymptotes of a hyperbola

The other day, I was reading up on the synthetic geometry of conic sections a bit, and I wondered: is it possible to construct the hyperbola's asymptotes given just the intersecting plane and the ...
0
votes
0answers
15 views

What is the analog of the scalar triple product in two dimensions?

Is there a standard name and/or a notation for the analog of the scalar triple product in two dimensions? Namely, i am interested in the following operation: given two elements $\vec u$ and $\vec v$ ...
0
votes
0answers
13 views

Circumcentre of three points X, Y, Z, given distance from each to points A and B

I'm racking my brain trying to figure out where to start on this, and it's been too many years since working on these kinds of problems. I have six measurements which I'd like to use to calculate a ...
0
votes
0answers
6 views

What is the proper name of a point a long a smooth curve where the radius changes but not direction of curvature?

What you call a point a long a smooth curve where the radius changes? When it reverses curvature, it’s an “inflection point”. What if it doesn’t change direction, just radius? I seem to remember ...
9
votes
4answers
336 views

Regular polygon inside another?

Inspired by this question, I was wondering if one can generalize to the case of an $n$-gon. For example, when $n=5$ we have this picture: where $ABCDE$ is a regular pentagon, ...
0
votes
1answer
40 views

euclidean distance between one dimension points-how to? [closed]

i am reading a research paper about round-robin scheduling algorithm that uses Euclidean distance to determine a time quantum based on similarity of burst times of all processes in the ready queue,but ...
3
votes
2answers
66 views

How long is the curve that a creature walks?

I have a problem in solving mathematical problem. Take a ball with radius 60 cm. A creature walk from the southpole to northpole by following the spiral curve that goes once around the ball every ...
0
votes
1answer
834 views

calculating volume of a horizontal cylindrical tank from depth

Has anyone found a good formula to convert a depth of fluid to a volume remaining in a tank for a cylindrical tank laying horizontal? (with or without half sphere end caps) Much Thanks!
1
vote
1answer
21 views

Pairs of isometries that jointly fix a set (revised)

Let $E$ be a Euclidean space, and let $X, Y\subseteq E$ be two disjoint subsets. Suppose that there exist isometries $i$ and $j$ such that $i(X)\cap j(Y)=\emptyset$ and $i(X)\cup j(Y)=X\cup Y$. Does ...
1
vote
0answers
52 views

Calculating Euclidean dissimilarity for a given cluster with itself

Suppose I have clusters $$A= \{(1,1)^T, (1,2)^T\} $$ $$B=\{(2,3)^T, (3,4)^T\} $$ $$C= \{(4,5)^T, (5,6)^T, (1,2)^T\} $$ I wish to use the Euclidean dissimilarity and Average linkage to calculate a ...
4
votes
2answers
115 views

Finding an angle between side and a segment from specified point inside an equilateral triangle

Here is the question: $\overset{\Delta}{ABC}$ is an equilateral triangle. D is a point inside triangle. $m(\widehat{BAD})=12^\circ$ $m(\widehat{DBA})=6^\circ$ $m(\widehat{ACD})=x=?$ I managed to ...
2
votes
1answer
601 views

Intersection of Two Circles

I have two circles as: $C_1: (x-x_1)^2+(y-y_1)^2=r_1^2$ and $C_2: (x-x_2)^2+(y-y_2)^2 =r_2^2$ and these circles have non-empty intersection. In other words $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\leq ...
3
votes
3answers
51 views

5 points on a plane with rational distances

Can you find 5 points on a plane whose Euclidean distances between them are all rational numbers and no 3 points out of them are co-linear? If the answer is yes, can we find a construction for ...
0
votes
1answer
13 views

Tranversal parallel lines theorem

I need to prove Tranversal parallel lines theorem that says: If two parallel lines are cut by a transversal, the corresponding angles are congruent, the alternate angles are congruent, and the ...
1
vote
1answer
221 views

Proof of a certain lemma in geometry

In the following article: http://yufeizhao.com/olympiad/geolemmas.pdf in the proof of the fact about the diameter of the incircle on page 2, the author claims that the proof that $BD = CF$ follows ...
2
votes
1answer
24 views

A triangular inequality including squares of sides

Show that for any triangle ABC, the following inequality is true $$a^2 + b^2 + c^2 > \sqrt{3} \max\{|a^2-b^2|,|b^2-c^2|,|c^2-a^2|\}$$ where $a,b,c$ are the sides of the triangle
1
vote
0answers
17 views

Inequalities in a quadrilateral

In a quadrilateral ABCD, it is given that AB is parallel to CD and the diagonals AC and BD are perpendicular to each other. Show that ...
0
votes
0answers
14 views

Central symmetry

There's definition: Central symmetry $s_O:M \to M$ is bijection defined as $s_O(T)=T'$ if and only if $O$ is midpoint of $\overline{TT'}$. 1st: Prove that central symmetry is involution ($s_O \circ ...
2
votes
1answer
22 views

How can I uniformly draw points from an ellipsoid?

Specifically, given a positive definite matrix $A \in \mathbb{R}^{n \times n}$, how can I efficiently generate points $x \in \mathbb{R}^n$ that satisfy $x^TAx \leq 1$? I know how to do this when the ...
5
votes
1answer
676 views

How to plot N points on the surface of a D-dimensional sphere roughly equidistant apart?

Let's say I have a D-dimensional sphere with a radius R. I want to plot N number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter where ...
1
vote
1answer
33 views

Nice geometry with areas

Consider triangle $ABC$. Let $H$ ortocentre and $O$ circumcentre, then prove that area of one of triangles $AOH$,$BOH$,$COH$ is equal to the sum of areas of other two.
1
vote
1answer
23 views

Plane-geometry problem with circles and tangents

I have a problem that even my smartest colleagues were able to solve. This is to get the radius of the smallest circle in the drawing below. Using a computer program, I managed to get that lightning ...
1
vote
2answers
27 views

Is this a Trapezium?

I once read that in hyperbolic geometry, two hyperbolas can be parallel. In a trapezium, you have four sides and a pair of parallel lines, therefore is it possible to have a trapezium with two ...
2
votes
1answer
70 views

Pairs of isometries that jointly fix a set.

Let $E$ be a Euclidean space, and let $X, Y\subseteq E$ be two disjoint subsets. Suppose that there exist isometries $i$ and $j$ such that $i(X)\cap j(Y)=\emptyset$ and $i(X)\cup j(Y)=X\cup Y$. Does ...
4
votes
1answer
28 views

Repeated projection of points onto lines

Consider a point $P$ on the Euclidean plane, and lines $l_1,l_2,\ldots,l_n$. Project $P$ onto $l_1$. Then project the resulting point onto $l_2$. Then project the resulting point onto $l_3$, and so ...
1
vote
2answers
24 views

On subgroups of isometries and their respective fixed-points

I am working on the following problemset: Let $G < \DeclareMathOperator{\Iso}{Iso}\Iso(E)$ be a finite subgroup of the isometries in the euclidean plane. Denote: $$G = \{g_1, \ldots , g_n \} ...
2
votes
0answers
12 views

Convexity of circle in neutral geometry

I am trying to prove that a circle is convex in neutral geometry. i.e. If $A$ and $B$ are inside a circle $C$, than any point in $AB$ is also in $C$. But I have difficulty in proving it. The case ...
2
votes
0answers
31 views

“Polysticks” in 3d

Consider a finite set of three-dimensional Euclidean vectors with integer components. How many three-dimensional closed loops can I construct with them? How many of them are elementary, i.e., cannot ...
2
votes
0answers
44 views

how to find angle between two added up vectors in cartesian space

I would like to find the angle between two vectors (theta) -> v1 From i to i+1 v1=(xi1-xi , yi1-y1) and v2 from i+1 to i+2 v2=(xi2-xi1, yi2-yi1), which are shown as in the figure (but v1 and v2 can be ...
0
votes
1answer
40 views

By proposition 3.21 is ether acute, right or obtuse

Let A, B, C be points on a line with A ∗ B ∗ C. Let D be a point not on that line such that angle ABD is obtuse. Show that angle CBD is acute. Do NOT use the Measurement Theorem. (Hint: By ...
2
votes
2answers
61 views

A geometry problem involving geometric mean

$ABCD$ is a quadrilateral inscribed in a circle of center $O$. Let $BD$ bisect $OC$ perpendicularly. $P$ is a point on the diagonal $AC$ such that $PC=OC$. $BP$ cuts $AD$ at $E$ and the circle ...
1
vote
1answer
30 views

Can these phenomena occur within Non-Euclidean geometries?

I've enrolled in an undergraduate seminar on the subject of non-euclidean geometry. I wanted to ground myself a little before-hand, because popular media has lead me to believe that non-euclidean ...
0
votes
1answer
26 views

Do collinear lines or overlapping collinear line segments intersect?

I am writing a function to find the intersection of a pair of lines and another function to find the intersection of a pair of line segments. The parallel case and the single point intersection case ...
0
votes
0answers
31 views

Distance on a 3-sphere

The arc-length $l$ between two points on on a 2-sphere of radius $R$ is given by $l=R\theta $ where $\theta$ is the subtended angle. I can rewrite this in terms of the euclidean distance $d$ between ...
1
vote
1answer
250 views

Convert Euclidean distance to Hyperbolic distance

I am searching for formulae to convert Euclidean distance into hyperbolic distance. The problem that I am confronting is that I want to calculate if a bug travels $x$ units in Euclidean space, how ...
0
votes
0answers
22 views

On power of the point

Let $P$ be a point outside the circle $O$ and $A$ a point on $O$ where $PA$ is tangent to $O$. Let $B, C$ be points on the $O$ such that $P$, $B$, and $C$ are collinear. Then $PA^2=PB\cdot PC$. Is ...
1
vote
3answers
42 views

Has triangle an angle?

I read axiomatic geometry and found the following definitions: Points $A$ and $B$ and all those points that lie between those points is a line segment. If $AB$ and $AC$ are two rays that does not ...
-2
votes
1answer
26 views

Neutral Geometry and Proclus' Axiom

Find a counter-example to Proclus' Axiom in the Klein disk. What can you conclude about Euclid's Parallel Postulate in the Klein disk?
2
votes
1answer
33 views

Barycentric Coordinates of the circumcenter of an arbitrary triangle

Given points $A(1, 0, 0), B(0, 1, 0), C(0, 0, 1)$ in barycentric coordinates, and points $P(x_P, y_P, z_P), Q(x_Q, y_Q, z_Q), R(x_R, y_R, z_R)$, what would the barycentric coordinates of the ...
0
votes
1answer
341 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
0
votes
1answer
22 views

Perimeter Of A Simple Triangle

Here in $ \triangle ABC$ $ AC=4 , DE= EF =1, \angle ABC=90^{\circ} $. The perimeter of the triangle $ \triangle ABC$ can be written as $ \sqrt {m } + n $ where $m$ and $n$ are non-negative ...
0
votes
1answer
26 views

Similarity. Finding distance.

Consider the figure. It is supposed to be a tennis court. A ball is served at $F$. It's trajectory is a straight line. The ball touches the ground at $A$. Find the distance $\mathbf x'$ from $A$ to ...
3
votes
2answers
89 views

The concurrence of angle bisector, median, and altitude in an acute triangle

$ABC$ is an acute triangle. The angle bisector $AD$, the median $BE$ and the altitude $CF$ are concurrent. Prove that angle $A$ is more than $45$ degrees. Here $D,E,F$ are points on $BC,CA,AB$ ...
1
vote
1answer
30 views

Distance of a point from a line

Let $w\in\mathbb{R}^{2}$ be a vector and denote $W=sp\{w\}$. Then $W^{\perp}$ is also a one dimensional space i.e is a line, denote this line as $l_{w}$. Given a line we can shift it from the origin ...
1
vote
1answer
34 views

Meaning of “circumference”

I am French and I have to solve a math problem written in English. The wording is the following : " In triangle ABC, the angle bisector of angle A intersects line BC at D and the circumference of ...
4
votes
1answer
26 views

Minimum dimension to hold $N$ points with given distances?

Suppose you're given $N$ points along with an $N\times N$ matrix $D$ with entries $d_{ij}$ giving the distances between the points (assume that the $d_{ij}$ satisfy the usual requirements of a ...