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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8
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174 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. (...
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1answer
31 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
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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
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1answer
35 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?
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1answer
19 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 ...
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4answers
56 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 $|AB|=|AC|...
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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 ...
14
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2answers
134 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, $$a^{-...
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1answer
33 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
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2answers
99 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
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1answer
24 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 $a,b,a',b'$ ...
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0answers
23 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 ...
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1answer
55 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 ...
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1answer
35 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?
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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...
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0answers
50 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 ...
2
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1answer
39 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
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1answer
25 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 $|\overrightarrow{p}|=2,|\overrightarrow{q}|=6,\angle(\...
0
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1answer
47 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 ...
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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 hypotenuse....
0
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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 $a$ ...
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1answer
33 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 ...
0
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2answers
139 views

Moving Line Segment Problem

So I was contemplating the following problem: Given two circles of radius $r$, which will position vertically so that their centers are separated by a distance of $2r+d$ (and thus, at the nearest ...
2
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1answer
103 views

condition for cones to be reciprocal

Question : Show that the cone $$ax^2 + by^2 + cz^2 - cxy - ayz - bzx = 0$$ is the reciprocal of the cone $$(a^2 - bc)x^2 + (b^2 - ac)y^2 + (c^2 - ab)z^2 - 2(a^2 + bc)yz - 2(b^2 + ac)zx - 2(c^2 + ab)xy ...
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1answer
20 views

Find $a\in\mathbb{C}$ so that the equation of the line through $−2 + i$ and $−2i$ is $z\bar{a} − a\bar{z} = 8i$

Consider the following geometry problem: Find $a\in\mathbb{C}$ so that the equation of the line through $−2 + i$ and $−2i$ is $$z\bar{a} − a\bar{z} = 8i$$ where the bar represents the complex ...
0
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1answer
32 views

Distance between two Polar-Coordinates

I choose two Points in Berlin with the coordinates: 1: lat: 52.511206 long: 13.546486 2: lat: 52.527501 long: 13.319206 With an online tool I got the ...
2
votes
4answers
69 views

Construct a triangle given certain lengths related to a bisector

Let $ABC$ be a triangle, and $AD$ the bisector of angle $A$. Write $AB = c$, $AC = b$, $AD = d$, $BD = c'$, $CD = b'$. Using ruler and compass, construct the triangle $ABC$ given the lengths $d$, $b-b'...
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1answer
60 views

Origami: What shapes are impossible?

Let's say we had a collection of pieces of rectangular paper of any size to choose from. Using one sheet only of any chosen size, what three-dimensional (or two-dimensional) shapes are impossible to ...
1
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1answer
58 views

Does this line preserving, continuous bijection on an equilateral triangle exist?

I'm trying to define a continuous bijection on the points (x/y coordinates) of an equilateral triangle. The vertices and the midpoints of each edge need to remain fixed. The (6) points on the edges ...
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14 views

Sphere overlap with cubic grid in R^N; minimal number of grid cells

I have: n-dimensional sphere of radius $r<0.5$ with position $\vec x $ generated randomly from uniform distribution $M$ different $N$-dimensional cubic grids, all with cell size $L=1.0$ which has ...
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1answer
40 views

Identifying the centre of a circle

I've been asked to identify with reasons of properties of circles , the centre of the circle which passes through Q1 . A, D and E Q2. A, C, And E . I'm not too sure how to approach this question
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2answers
50 views

How to find whether a point lies on a line which is in parametric form?

Does the point $(1,8,3)$ line on the line with parametric equation: $$x = 5 + 2t$$ $$y = 2 + 6t$$ $$z = 1 + 3t$$ I know how to solve if they give me a equation of a plane and ask whether the ...
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2answers
55 views

Properties of circles

I'm told to find angle CAD and I got stuck... Given that CD = BC My workings .. CAT = ACT = $180-40/2 = 70$ CAP = $180-70-60=50$
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1answer
22 views

Finite points in geometry

What are finite points in Geometry? I was reading this and they mention "Suppose that $P$ and $U$ are finite points having normalized barycentric coordinates $(p,q,r)$ and $(u,v,w)$." I am wondering ...
2
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0answers
79 views

A chain of six circles associated with a conic

I am looking for a solution of the following problem: Let $A_1, ..., A_6$ and $B_1, ..., B_6$ be 12 points lying on a conic, and suppose that for $i=1, ..., 5$ through $A_i, A_{i+1}, B_{i+1}, B_i$ ...
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3answers
143 views

Construct a parallelogram subject to certain conditions

I am having trouble with the following exercise from Dollon and Gilet's Géométrie plane. Two parallel lines $\Delta$ and $\Delta'$ are given, as well as a point $A$ on $\Delta$ and a point $O$ on ...
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1answer
46 views

Finding angle associated with point inside an equilateral triangle.

$\triangle{ABC}$ is an equilateral triangle. $|AD|=6$. $|BD|=10$. $|CD|=8$. What is $m\angle{CDA}$? First thing comes to mind is Ceva theorem. I used its trigonometric form to reach ...
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1answer
31 views

Intersection of line and plane [closed]

I'm trying to figure out a problem in perspective geometry. Given the line from the origin to the point $P(x,y,z)$ at some point in space it casts a picture of a point when intersecting the plane $y=1$...
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1answer
26 views

Area of quadrilateral with equilateral triangles erected on sides (

Let $ABC$ be a triangle. Construct points $B'$ and $C'$ such that $ACB'$ and $ABC'$ are equilateral triangles that have no overlap with $\triangle ABC$. Let $BB'$ and $CC'$ intersect at $X$. If $AX = ...
3
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4answers
63 views

Definition of an angle

I saw two definitions of an angle. Are those equivalent or is another wrong in some axiomatic system? An angle is the union of two rays. An angle is a subset of a plane restricted by two rays. I ...
2
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0answers
26 views

How to construct a polyhedron from given planes

This seems to be a basic questions, but I really don't know a good computer algorithm to do this. I have a set of planes (parameterized by normal direction and distance from a given point), and I want ...
0
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1answer
30 views

4 known points, 8 unknown - problem of uniqueness of solution

I have 4 known points $\textbf{P}_i \quad (i = 1, 2, 3, 4)$. One of these points is simply $\textbf0$. I have 8 unknown points: $\textbf{F}_j$ and $\textbf{R}_{jk} \quad (j = 1,2; \quad k = 0,1,2)$. ...
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1answer
35 views

What kind of line is formed when a piece of paper is folded?

What kind of line is formed when a piece of paper is folded? I am 17, and I'm homeschooled. I have a book from 1958 that's called "Plane Geometry, Welchons Krickenberger Pearson" It has no answers in ...
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2answers
208 views

Geometrical proof required regarding midpoints

$ABCD$ is a convex quadrilateral with $W, X, Y, Z, M$ and $N$ as the midpoints of $AB, BC, CD, DA,$ the diagonals $AC$ and $BD$ respectively. [Then, $WXYZ$ is a parallelogram with $K$ as the ...
3
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2answers
45 views

How to prove that this is a parellelogram?

Prove that ABED is a parallelogram Given: ABCD is a trapezium F and G are the midpoints of AB and DC respectively FHG is a straight line AD is equal to and parallel to BE My attempts have ...
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1answer
59 views

Trisecting an an angle - how to prove?

I have this question on my "History of Mathematics" problem set: Draw any angle AÔB; Pick a point C on OB; Now trace CD which is perpendicular to OA; Draw a parallel line $s$ to OA which passes ...
2
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2answers
67 views

Three Altitudes of a triangle are concurrent

I have been told that this well known fact can be shown using only Euclid's propositions from books one to three, and cyclic quadrilaterals. I can't figure out how to start, which quadrilateral ...
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2answers
48 views

Distance inequality

Consider $H=\{ (x,y) \in {\bf R}^2\mid x$ or $y$ is an integer $\}$ If $d$ is canonical distance in ${\bf R}^2$, show that if $d(x):=d(x,H)$, (1) $$ d(x) - d(y) \leq d(x,y) $$ if $x,\ y$ are in same ...
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1answer
15 views

Show that if $Q'$ is any point on the line of action of $F$, then $PQ × F$ = $PQ'× F$

If a force $F$ is applied to an object at a point $Q$, then the line through $Q$ parallel to $F$ is called the line of action of the force. We defined the vector moment of $F$ about a point $P$ to be $...
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2answers
37 views

Area enclosed by a polygon

I did some work in the area of mensuration and came across an interesting concept/formula. The formula states that For a polygon having vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3),\dots(x_n,y_n)$, the ...