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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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 ...
2
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2answers
41 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 $h,k$. ...
14
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3answers
832 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
43 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: ...
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0answers
24 views

tessellation of an arbitrary shape [on hold]

Are there any shapes that we can tessellate any plane shapes (with arbitrary shapes) by them? i.e. if I generate a random shape, how can I tessellate it by some shapes?
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1answer
17 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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1answer
38 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 $DE$ (extended if necessary), meet the perpendicular from $C$ on $AB$ in ...
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0answers
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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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1answer
23 views

The limit incentral triangle is equilateral [on hold]

I found a nice problem of geometry but I don't know how to prove it. Given a triangle $T_0$, we build $T_1$ by considering the projections of the incenter of $T_0$ on the sides of $T_0$. In the ...
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0answers
17 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 ...
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0answers
15 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 ...
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1answer
34 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 ...
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1answer
33 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: ...
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1answer
41 views

Formula of the radius of a circle [on hold]

What is the formula of the radius of a circle circumscribing by the quadrilateral?
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1answer
7 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
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0answers
62 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 ...
12
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1answer
199 views
+100

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
14 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
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0answers
22 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
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2answers
78 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), ...
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0answers
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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 ...
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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 ...
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2answers
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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
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1answer
49 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 ...
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2answers
48 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 ...
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0answers
37 views

converting 2d points on an picture to a 3d plane

I want to ask the same question as from this tread, about planar homograph but using absolute values so I can visualize it. My goal is to retrieve 3D transformations of a plane (position/rotation ...
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0answers
11 views

Discrete grid: random points with radial probability distribution

I have a cubic 3D grid of $N^3$ points. I randomly choose a certain point to be the centre. Now I want to generate random points which obey a certain probability distribution $\rho(r)$ which depends ...
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0answers
44 views

Given x,y,w,h can you generate a rainbow box/cuboid with rounded edges?

Given $x$, $y$, $w$, $h$ where $0 \leq x < w$ and $0 \leq y < h$ and $(x, y)=(0, 0)$ is bottom-left and $(x, y)=(w-1, h-1)$ is top-right and they're all integers, can you make a formula that ...
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1answer
34 views

Two circles intersect in two points and the line through these two points

Consider two circles $C,C'$ in euclidean plane which intersect in exactly two points $Q,R$ and consider the line $QR$ through these points. The claim is that a point point $P$ lies on the line $QR$ ...
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Alternative proof for the equality of two angles in an isosceles triangle.

From the answers of my previous question, I got an idea to prove equality of two angles in an isosceles triangle. In that question the equality of two angles in a right-angled-isosceles triangle was ...
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1answer
10 views

Equation of a line about which we are reflecting

Let $A$ be the matrix of a reflection about a line of the euclidean plane (w.r.t. the standard basis). How can I find the equation of the line?
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3answers
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Reflections in Euclidean plane

Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be the counterclockwise rotation of $\frac{\pi}{2}$ and $S: \mathbb{R}^2 \to \mathbb{R}^2$ be the reflection w.r.t. the line $x+3y=0$. There exists a reflection ...
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On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.

So, according to this figure : http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII23.html We cannot have similar segments of circles and unequal ones be built on the same side of the same ...
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1answer
59 views

Points on a 2D plane spanned by a turtle graphics system

Suppose you have a turtle graphics system with a set "turning angle" $\delta$, in which the turtle can execute three commands: $F$: Go forward, by unit length, in the current direction. (The initial ...
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1answer
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In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base.

I have the following theorem : "In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base." (Figure is in the link) ...
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3answers
50 views

How can I find the diagonal of a quadrilateral? [on hold]

Given a quadrilateral $MNPQ$ for which $MN=26$, $NP=30$, $PQ=17$, $QM=25$ and $MP=28$ how do I find the length of $NQ$?
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0answers
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Horn angles and Euclid's elements.

We have the following statement by Euclid : "I say further that the angle of the semicircle contained by the straight line BA and the circumference CHA is greater than any acute rectilinear angle, and ...
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1answer
33 views

Euclid's elements proposition 15 book 3

http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII15.html I have understood the proof in general. It is only a small detail which i'm not sure. Maybe it's because english isn't my first ...
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1answer
32 views

Euclid's elements proposition 13 book 3

"Then, since on the circumference of each of the circles ABDC and ACK two points A and C have been taken at random, the straight line joining the points falls within each circle, but it fell within ...
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1answer
70 views

Why is the Euclidian norm used to measure complex numbers?

Why is the Euclidian norm used to measure complex numbers? The complex numbers are numbers (or more precisely, pairs of numbers), and I can't see why are they essentially connected to the ...
1
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1answer
38 views

How many sets of four points in an MxN grid have one point contained by three other points?

Given a 3x3 grid: 1 2 3 8 9 4 7 6 5 We find 126 distinct sets of 4 points $$\binom{9}{4}$$ There are 8 cases such that when the points are connected with a line in clockwise direction, one point ...
3
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1answer
58 views

Euclidean triangle is determined by Angle, Median and radius of exterior circle

consider two triangles $\Delta(A,B,C), \Delta(A',B',C')$ in euclidean plane. I want to prove that these triangles are congruent if they are equal in the following data: they have the same angle at ...
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1answer
36 views

Online tool for making Geometric Constructions.

There was a website where it tasked you making different geometric shapes using only a compass and straightedge. I've looked for it and I can't find it or even discussion about it. What I do remember ...
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1answer
28 views

Lengths on the unit octahedron

Consider the face of the unit octahedron, defined by: $$O^2 = \{(x,y,z): |x|+|y|+|z|=1\}$$ Every point on the octahedron has between 0 and 3 positive coordinates. E.g, in $(0.2,-0.3,0.5)$, the $x$ ...
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2answers
57 views

Why is this point set a circle?

consider a circle in Euclidean plane $E$ and any point $A$ in the interior of the circle. Now consider all secants $s_A$ to the circle through the point $A$. The claim is now that the set of midpoints ...
2
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2answers
24 views

Proof of folding to trisect a right angle

If first you fold a normal (letter or A4) piece of paper in half: and then you fold one corner to meet the halfway line: Then you've trisected the right angle at bottom left - but how does one ...
1
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1answer
59 views

Find my coordinates from distance with unknown coordinates

I am trying to work out if there is a way to calculate some coordinates relative to each other simply by knowing $3$ or more distances from some unknown points. I do not have a distance matrix, I ...
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2answers
52 views

Show that $\triangle ABK \cong \triangle ABL$. [closed]

Show that $\triangle ABK \cong \triangle ABL$ where $D$ is the circumcenter,$K$ is the orthocenter and L lies on the circle.
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19 views

Solid angle in $D$ dimensions

Consider $d\Omega_D$, the element of solid angle in dimension $D$. Suppose an integrand depends only on $m-1$ of the angles (so that it doesn't depend on the set $\left\{\phi_i, i=1,\dots,D-m\right\}$ ...