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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51 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
37 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
11 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
92 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 ...
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1answer
302 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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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 ...
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24 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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10 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 ...
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2answers
91 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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141 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 ...
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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
22 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 ...
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1answer
59 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
94 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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51 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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13 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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50 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
45 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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54 views

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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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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26 views

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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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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25 views

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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51 views

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 $NQ$?
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27 views

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
35 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
36 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
73 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 ...
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1answer
45 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 ...
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63 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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53 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
29 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
59 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 ...
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2answers
34 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 ...
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1answer
70 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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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 $\left\{\phi_i, i=1,\dots,D-m\right\}$ ...
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About pythagorean triples

In the circle of diameter $AB$ it is well known each point $C$ determines a right triangle $\Delta ABC$ and so it is with every point $D$ on the circle of diameter $AC$ determining a right triangle ...
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How do I calculate the angles between a point on a sphere and each unit vector in $\Bbb R ^3$?

Given the Cartesian coordinates of any point $p$ on the surface of a sphere in $\Bbb R ^3$, how do I calculate the angles between each axis $(x, y, z)$ and the vector $n$ defined by origin $o$ and ...
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2answers
53 views

any good sourcebook for plane geometry problems?

I wanted to find some good resource books for euclidean plane geometry. would anybody name some titles?
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2answers
110 views

What's interesting in latus rectum?

I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical ...
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Faster Alternative than Calculating Euclidian Distance to determine which Coordinate has Max Distance from a fixed coordinate (eg (0,0))

I am developing a program that needs me to determine which coordinate in a 2-d figure has maximum distance from a fixed coordinate. Let me demonstrate: 3 points: (1,3), (2,2), (5,0) ; Fixed point: ...
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Construct a matrix satisfying a linear restriction with bounded singular values

Suppose for some matrix $A\in\mathbb{R}^{n\times n}$, and some vectors $x,y\in\mathbb{R}^n$, $y=Ax$. Under what conditions (ideally on $A$) can one construct a matrix $B\in\mathbb{R}^{n\times n}$ ...
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1answer
27 views

Straightedge-Only for Perpendicularity

Given a triangle ABC and a midpoint M (of the line AB), is it possible to check whether the line CM is perpendicular to AB with a straightedge only? By this, I mean that points can be added ...
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1answer
26 views

Finding distance from point to line which is perpendicular to another line

Find the distance of the point $(1,1,1)$ from $x+y+z=1$ measured perpendicular to the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{6}$
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Is there a term for two distance metrics that give the same ordering?

In order to calculate and sort things by distance from each other in a computer program I need to find an easy and fast distance metric to calculate. I can probably find one myself. However, first I ...
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1answer
50 views

Surface Area of unit n-sphere covered by rotating a unit vector around a fixed unit vector such that angle between the two vectors is always fixed.

Consider an n-dimensional unit sphere and unit vector from the origin with its tip lying on the surface of sphere. Consider another vector which makes some angle say $\epsilon$ with unit vector. From ...
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1answer
74 views

Moving circular disk between two parallel sinusoidal curves

Find the largest radius of the circle that can be "rolled" between the curves $y = sin(x)$ and $y = sin(x)+1$. After two weeks of research, I finally give up.
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53 views

Proof that an angle across line is equal to 180 degrees

When given a straight line, how do you prove that an angle across it is equal to 180 degrees, or two right angles? It feels like something that should be an axiom, but it isn't one of the 5 ...