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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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 ...
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0answers
19 views

About Homothetic transformation

I have one question regarding the way which a homothetic transformatior is written. Why is it written in the following way: $$\vec{OH^{k}_{O}(P)}=k\vec{OP}+(1-k)\vec{OO}?$$ From where ...
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25 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. ...
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3answers
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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 $a$ ...
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2answers
22 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 ...
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1answer
24 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 ...
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2answers
39 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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1answer
43 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
27 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
20 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
59 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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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 ...
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5answers
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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
25 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 ...
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2answers
63 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 $h,k$. ...
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4answers
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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
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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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tessellation of an arbitrary shape [closed]

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
19 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
41 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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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
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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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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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1answer
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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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36 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 [closed]

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 ...
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1answer
83 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
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1answer
286 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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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 ...
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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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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 ...
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2answers
81 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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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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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 ...
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1answer
50 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
50 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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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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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
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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
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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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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
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How can I find the diagonal of a quadrilateral? [closed]

Given a quadrilateral $MNPQ$ for which $MN=26$, $NP=30$, $PQ=17$, $QM=25$ and $MP=28$ how do I find the length of $NQ$?