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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2
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1answer
54 views

Stuck on computing distance travelled from velocity and yaw rate.

I am somewhat stymied on what appears to be a simple formula. Here is the problem statement: Assume that a rigid body is traveling with constant velocity $v$, and is rotating with a constant yaw rate ...
2
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2answers
60 views

Trying to understand the limit of regular polygons: circle vs apeirogon (vs infinigon?)

In the definition of regular polygon at the Wikipedia, there is this statement about the limit of a n-gon: "In the limit, a sequence of regular polygons with an increasing number of sides becomes ...
3
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2answers
38 views

How is a vertex of a triangle moving while another vertex is moving on its angle bisector?

While trying to solve this problem the following conjecture came to my mind. Based on the statement conjectured I could solve the problem mentioned. I am unable to verify the statement that I found ...
3
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1answer
49 views

Construction of a triangle with given angle bisectors [duplicate]

given three distinct lines $g,h,l$ meeting in one point $P$. I want to construct a triangle with vertices on $g,h,l$ such that those lines $g,h,l$ become its angle bisectors. In general, if we ...
0
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1answer
19 views

How do you find the composition of three functions

We have learned that if you let $P$ be the Euclidean plane with distance $d$, a function $F: P \to P$ is an isometry if, for all points $X$ and $Y$ of $P$, $d(F(X),F(Y)) = d(X,Y)$. Also the following ...
12
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1answer
81 views

Uniqueness of a configuration of $7$ points in $\Bbb R^2$ such that, given any $3$, $2$ of them are $1$ unit apart

This question from earlier today asks (paraphrasing here): Is there a configuration of $7$ points in the Euclidean plane such that, given any $3$ of the $7$ points, at least $2$ of them are $1$ ...
3
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2answers
82 views

Curvature flow for convex planes curves

Tentative translation of the original question. I've read several articles on the curvature flow for convex plane curves (the curve remains convex during evolution, and eventually shrinks to a point). ...
0
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2answers
52 views

Find the area of a triangle whose vertices cut the sides of $ABC$ in thirds [closed]

I have to find the area of $F$, given the following configuration: $\hspace1in$ What to do?
3
votes
2answers
89 views

Seven points in the plane such that, among any three, two are a distance $1$ apart

Is there a set of seven points in the plane such that, among any three of these points, there are two, $P, R$, which are distance $1$ apart?
2
votes
1answer
53 views

Construct with straight edge a parallel to two lines.

It is known that we can't with just a straight edge, given a line and a point out of the line in a plane to construct other line, passing through the point, parallel to the first. I know a proof of ...
1
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2answers
34 views

Existence of Linear Transformation between 3D line and 2D line

I am wondering if there exists an invertible linear transformation between a line segment in 3D space and a line segment in 2D space. Basically, the red line above could be represented by the ...
6
votes
2answers
56 views

Subsets of a set of points that can lie in the same sphere

Suppose I have a finite set $\mathcal{P} := \{x_1, x_2, \ldots , x_n\} \subset \mathbb{R}^d$. Is there any way to characterize the couples $(x_i, x_j)$ such that there exists a ball $B$ with $x_i, ...
0
votes
1answer
16 views

Sweep-Line Algorithm complixity

Sweep Line Alg:- is a kind of algorithm that uses a conceptual sweep line or sweep surface to solve various problems/issues in Euclidean space geometric. It is one of the key techniques in ...
0
votes
0answers
14 views

An integral from the integral geometry about the isoperimetric inequality.

The problem is from the book "Integral Geometry and Geometric Probability" by Santalo (1976), Chapter 1.3.5, Notes and Exercises (page 37). Given a convex closed curve $C$. Let $A_1$, $A_2$ be the ...
1
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1answer
22 views

What's the name given to the ratio $P^2/A$ for a closed figure in the Euclidean plane?

Let $\mathscr{F}$ be the set of all plane, closed Euclidean figures having positive perimeter, and let $\sim$ be the similarity relation on $\mathscr{F}$. Then, for any equivalence class ...
0
votes
1answer
44 views

Prove base times altitude is a constant without resorting to area.

The area of a triangle is one-half base times altitude. This implies that, for $\bigtriangleup ABC$, $ah_A = bh_B=ch_C$, where $h_A$ is the length of the altitude dropped from point A to side BC, etc. ...
1
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3answers
65 views

Find the area of triangle, given an angle and the length of the segments cut by the projection of the incenter on the opposite side.

In a triangle $ABC$, one of the angles (say $\widehat{C}$) equals $60^\circ$. Given that the incircle touches the opposite side ($AB$) in a point that splits it in two segments having length $a$ ...
0
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2answers
29 views

Does being distance-preserving force affinity?

Let $\phi: \mathcal{E}\to \mathcal{E}$ be a mapping defined on a Euclidean affine space. Assume that $\phi$ preserves the distances: $$d(\phi(A),\phi(B)) = d(A,B).$$ Prove that $\phi$ is affine, and ...
0
votes
1answer
31 views

Why is not parity transformation just a rotation?

I'm a bit confused about parity transformations (reflections). A parity operator $\pi$ takes a vector $(x, y, z)$ to $(-x, -y, -z)$. So in a $3$ dimensional space this takes a vector and points it ...
-1
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2answers
45 views

Question about the existence of points and lines.

Say we draw a point on a graph. If the point should not take up any area than how come we could see it. Say we graph $y=x^2$, we obviously could see it. However, because $y=x^2$ is a function made up ...
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votes
1answer
50 views

Proving Ordering of Angles

I'm trying to prove $$\text {If}\ \angle P \lt \angle Q, \text & \ \angle Q \cong \angle R, \text{Then}\ \angle P \lt \angle R$$Which seems super basic and makes sense, but I got told that I'm ...
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2answers
37 views

Find 3D distance between two parallel lines in simple way

Is there a simple way to get 3D distance between two parallel lines given end points of each line?
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1answer
27 views

The projective space of all lines through the origin

I have a question to the following example: Assume that $\mathbb{A}_2$ is an affine plane over a field $\mathbb{K}$, and we have fixed affine coordinates $x, y$ on $\mathbb{A}_2$. Let $\mathbb{P}$ be ...
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2answers
73 views

All right angles are equal to each other

Why is it that All right angles are equal to each other -a postulate in Euclid's Elements (Wikipedia). Shouldn't it be a congruence rather than an equivalence? Isn't this just a special case of ...
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0answers
35 views

Pascal's theorem in ellipse

I know that the proof of Pascal's theorem in circle is based on the power of a point. I also heard that one can use projective geometry to prove it in an ellipse. But is there a direct synthetic proof ...
3
votes
5answers
52 views

Showing that certain points lie on an ellipse

I have the equation $$r(\phi) = \frac{es}{1-e \cos{\phi}}$$ with $e,s>0$, $e<1$ and want to show that the points $$ \begin{pmatrix}x(\phi)\\y(\phi)\end{pmatrix} = ...
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2answers
27 views

Incommensurable line segments

I have an issue with a proof given in my lecture hopefully someone can help me with. It will be shown that the hypotenuse $c$ of a right-angled and isosceles triangle will be incommensurable to the ...
3
votes
2answers
64 views

Construction of a circle through a point and tangent to angle

given an angle $\angle (h,k)$, where $h,k$ are the legs of the angle. Let $P$ be some point in the interior of the angle. I want to construct a circle through P which is tangent to both legs $h,k$. ...
16
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4answers
1k views

The position of a ladder leaning against a wall and touching a box under it

I was reading a newspaper and there was a little math riddle, I thought "how funny, that's gonna be easy, let's do it" and here am I... The problem goes as follow : in a barn, there is a 1 meter ...
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3answers
49 views

Is it possible to find the vertices of an equilateral triangle given its center point?

I was wondering how to find the vertices of an equilateral triangle given its center point? Such as: ...
1
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1answer
26 views

Points on two skew lines closest to one another

Given two skew lines defined by 2 points lying on them as $(\vec{x}_1,\vec{x}_2)$ and $(\vec{x}_3,\vec{x}_4)$. What are the vectors for the two points on the corrwsponding lines, distance between ...
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votes
3answers
63 views

Two prove two lines in a triangle are parallel

$D$, $E$, $F$ are the midpoints of sides $BC$, $CA$ and $AB$ respectively of a triangle $ABC$ right angled at $C$. If $EF$ and $DF$ (extended if necessary), meet the perpendicular from $C$ on $AB$ in ...
2
votes
1answer
43 views

To prove two angles are equal when some angles are supplementary in a parallelogram

The point 'P' is situated inside the parallelogram ABCD such that the angles APB and CPD are supplementary.Prove that angles PBC and PDC are equal
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0answers
21 views

Dropping parallel postulate and infinitude of straight line

I was reading an article about back ground of Killing's work by Thomas Hawkins from Historia mathematica 1980.It was written that Killing stated that if one drop assumptions infinitude of straight ...
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0answers
22 views

To circumscribe a square about a given circle.

http://aleph0.clarku.edu/~djoyce/java/elements/bookIV/propIV7.html I was just wondering something . We know that if a line touches a circle at one point, then this means that this line is forming a ...
1
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1answer
35 views

Equivalent definitions of symmetry group of regular n-gon (dihedral group)

Let $P_n$ be a fixed regular convex $n$-gon in the plane. For a metric space $M$ we denote by $\text{Isom}(M)$ the set of distance-preserving maps $M \to M$. How can I show that $$ D_n := \left\{\, f ...
0
votes
1answer
48 views

Construct the triangle with given angle bisectors

given three lines $\ell_1,\ell_2, \ell_3 $ which intersect in one point $P$. How can one construct a triangle such that the given lines become its angle bisectors? So far I tried to find conditions ...
0
votes
1answer
35 views

Construction of triangle from side $c$ and heights $h_a, h_b$

I want to construct a triangle $\Delta(A,B,C)$ with given side $c$ and heights $h_a, h_b$. To construct the triangle means to use only ruler and compass. How can I solve this? I started as follows: ...
0
votes
1answer
9 views

I am looking for a function in order to measure points misalignment

The points are in the euclidean plane, let $\mathbb{P}$ be the set containing all the finite sets of $\mathbb{R}^2$ points. I am looking for a function $m : \mathbb{P} \to \mathbb{R}^+$ in order to ...
2
votes
1answer
89 views

Find the Height of the Trapezoid

Problem: The area of a trapezoid is equal to 2 and the sum of his diagonals is equal to 4. Find the trapezoid height. [QUESTION]: I find a result that implies that the height of the ...
13
votes
1answer
294 views

How big is my pizza, if I know its slices' sizes?

I bought a box of frozen pizza: eight slices, baked and then frozen, stacked in a box. The packaging assured me that I was holding an 18-inch[-diameter] pizza. That got me thinking: how do I know ...
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0answers
15 views

Planar nearest neighbor search for many points.

I have two sets of points on the plane, A and B. For every point in A, I would like the k nearest points in B. The naive approach is for each point in A having a linear selection to choose the kth ...
2
votes
0answers
23 views

Name of the segment connecting a point's coordinate axis projections?

Given any point $(x,y)$ in the real plane consider the corresponding line segment connecting $(x,0)$ with $(0,y)$. See diagram. Is there a name for this special segment? (I believe that in ...
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0answers
8 views

Identifying Peak Points in a 3-dimensional space

I have a data set which is composed of $xyz$ points. I want to be able to identify peak and valley points from the set. Identifying peak for 2D points $(x,y)$ is simple. I want to know what properties ...
3
votes
2answers
86 views

Let $ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$. Determine $S+S+…+S $.

Let $$ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$$ By the usual notation for sum of sets let $$ 2S\overset{\text{not}}{=}S+S=\{(x_1+x_2,y_1+y_2) \ | \ (x_1,y_1), ...
5
votes
1answer
132 views

Space Geometry: lines in a plane

If $d$ and $d'$ are two intersecting lines in a plane $P$, and $D$ is a line orthogonal to both $d$ and $d'$, then any line $\delta$ in $P$ is orthogonal to $D$ as well. How could this be proven ...
0
votes
0answers
21 views

What's the operator, standard matrix and effect on the standard basis for the shear transformation in three dimensions?

I know that the operator for shear in $\mathbb{R}^2$ in the $x$-axis is $T(x,y)=(x+ky,y)$. And on the $y$-axis it is $T(x,y)=(x,y+kx)$. But what about in three dimensions? Is it something like ...
2
votes
2answers
20 views

Construct non trivial group endomorphism (rigid motion group)

My question, in it's general formulation, is : is there a way to construct non trivial group endomorphism other than conjugation ? Now for my specific needs, I wont to find some endomorphism other ...
3
votes
1answer
53 views

How to find the center of a log spiral?

Given just a few points on a log spiral, how to find the center? Considering that the circle is a degenerate case of the log spiral, is there a way to generalize the method for finding circle centers ...
0
votes
2answers
63 views

Weighted Average Distance between 3 or more points

I'm trying to find out the average point between 3 or more points, given each distance that the end point is away from each of the other points. With 2 points it's easy, I believe the formula should ...