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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On which side of vector the point is

I have a vector A and a point P. My problem is how to find out if point is on the left or right side of vector looking from the point of origin of a vector in direction of it.
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4answers
8k views

How to find a point on a line closest to another given point?

Given the line $x+2=\frac{y+4}{2}=\frac{z-5}{-2}$ I want to find the closest point on this line to $(1,1,1)$ I suppose the details here don't matter but in general how is this done? We need a ...
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2answers
226 views

Analytic Geometry | Two Planes and a Angle | Two Solutions

This is me again, I have another problem which I haven't been able to solve, the legend goes like this: Find the equation of the plane that contains the points $P_1(1,0,-1)$, $P_2(2,0,2)$ and forms a ...
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1answer
190 views

Can we alter Hilbert's axioms to have $\mathbb{Q}^3$ as a unique model?

Can we alter Hilbert's axioms to have $\mathbb{Q}^3$ as a unique model? The critical axioms seem to be the congruence axioms IV.1 and IV.4, and presumably the line completeness axiom V.2. But how ...
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0answers
168 views

Characterizations of Euclidean space

There are presumably three ways of characterizing the abstract Euclidean space $E^n$ that are quite different in spirit: axiomatically (with axioms concerning dimension) by the abstract Euclidean ...
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0answers
255 views

Euclidean geometry and the Euclidean group

At Wikipedia's Erlangen program I read that "quite often, it appears there are two or more distinct geometries with isomorphic automorphism groups". Some examples are given. But what are examples ...
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1answer
300 views

Curve of a fixed point of a conic compelled to pass through 2 points

Suppose that in the plane a given conic curve is compelled to pass through two fixed points of that plane. What are the curves covered by a fixed point of the conic, its center (for an ellipse), its ...
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1answer
246 views

How to compute the change in the angle between two unit norm vectors as the $\ell_1$ norm of one vector changes?

Motivation Suppose that $u \in \mathbb{R}^d$ is a unit-norm vector, $\|u\| = 1$, $a, b, c$ are some positive constants and $\xi \in [0,1]$ is another constant (usually chosen close to 1). I am ...
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5answers
261 views

What type of triangle satisfies: $\cot \biggl( \frac{A}{2} \biggr) = \frac{b+c}{a} $

If in a $\displaystyle\bigtriangleup$ ABC, $\displaystyle\cot \biggl( \frac{A}{2} \biggr) = \frac{b+c}{a} $, then $\displaystyle\bigtriangleup$ ABC is of which type ?
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2answers
3k views

How to calculate area of triangle having its points 2D coordinates?

We have points A, B & C in 2D plane. How having point coordinates $(x, y)$ to calculate area of triangle formed by them?
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2answers
4k views

Euler angles and gimbal lock

Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading ...
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5answers
995 views

Why the interest in locally Euclidean spaces?

A lot of mathematics as far as I know is interested in the study of Euclidean and locally Euclidean spaces (manifolds). What is the special feature of Euclidean spaces that makes them interesting? ...
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5answers
208 views

Existence of lines not containing given points in general position

I remember seeing something like the following problem in the past and would like to know if it has a solution (or if I can find a source for it). Problem Given a finite set of points in the plane in ...
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8answers
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How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?

I would like to find the apothem of a regular pentagon. It follows from $$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$ But how can this be proved (geometrically or trigonometrically)?
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1answer
583 views

Has anything further been done with Morley’s Miracle?

Or has it remained a terminal node at the frontier of mathematics?
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474 views

Decomposing the plane into intervals

A recent Missouri State problem stated that it is easy to decompose the plane into half-open intervals and asked us to do so with intervals pointing in every direction. That got me trying to ...
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2answers
2k views

What is wrong in my proof that 90 = 95? Or is it correct?

Hi I have just found the proof that 90 equals 95 and was wondering if I have made some mistake. If so, which step in my proof is not true? Definitions: 1. $\angle ABC=90^{\circ}$ 2. $\angle ...
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3answers
214 views

Find all points with a distance less than d to a (potentially not convex) polygon

I have a polygon P, that may or may not be convex. Is there an algorithm that will enable me to find the collection of points A that are at a distance less than d from P? Is A in turn always a ...
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4answers
264 views

Must Basis of an Euclidean Space Be Ordered

Does the basis of an Euclidean space have to be ordered by definition? Or can be left unordered? I was also wondering about what is the morphism (i.e. the mapping that can preserve all the ...
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3answers
448 views

How can I remove rotations from points defining a plane?

I have coordinates for 4 vertices/points that define a plane and the normal/perpendicular. The plane has an arbitrary rotation applied to it. How can I 'un-rotate'/translate the points so that the ...
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3answers
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Proof that every polygon with an inscribed circle is convex?

In many elementary (and not-so-elementary) Euclidean geometry texts, a (simple) polygon is said to be tangential  if it is convex and has an inscribed circle (i.e., a circle that intersects and ...
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2answers
387 views

Does the nine point circle generalise to some theorem about n-spheres and n-simplices?

I am obsessed with the nine point circle. I was thinking, is there a generalisation to aribtrary tetrahedra and spheres? What about higher dimensions? For each face of the tetrahedron, there is a nine ...
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1answer
581 views

How can I pack $45-45-90$ triangles inside an arbitrary shape ?

If I have an arbitrary shape, I would like to fill it only with $45-45-90$ triangles. The aim is to get a Tangram look, so it's related to this question. Starting with $45-45-90$ triangles would be ...
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2answers
837 views

Deriving volume of parallelepiped as a function of edge lengths and angles between the edges

In Wikipedia it is stated that the volume of the parallelepiped given its edge lengths $a,b,c$, and the internal angles between the edges $\alpha ,\beta ,\gamma $ is: \begin{equation*} ...
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2answers
541 views

Computing the moments of a triangle

Assume you have a triangle in the plane defined by its three vertices at $(x_0,y_0)$, $(x_1,y_1)$, and $(x_2,y_2)$. Is there a general expression for the moments of the triangle, where the moments are ...
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2answers
737 views

Rotations by degrees other than $90, 180,$ and $270$.

Say I have a triangle with vertices $(0,0), (2,4), (4,0)$ that I want to rotate along the origin. Rotation by multiples of $90^{\circ}$ is simple. However, I want to rotate by something a bit more ...
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6answers
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What is the Direction of a Zero (Null) Vector?

To be more precise, I am interested in knowing if the intuition that a Euclidean zero vector does not have a particular direction is actually correct, and if there is a rigorous formulation that would ...
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3answers
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Minimal Ellipse Circumscribing A Right Triangle

Find the equation of the ellipse circumscribing a right triangle whose lengths of it's sides are $3,4,5$ and such that its area is the minimum possible one. You may chose the origin and orientation ...
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6answers
2k views

Where does the Pythagorean theorem “fit” within modern mathematics?

I am interested in how today's professional mathematicians view the Pythagorean theorem, in terms of how the theorem fits within the axiomatic framework of mathematics. I often come across textbooks ...
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3answers
4k views

Find the coordinates in an isosceles triangle

Given: $A = (0,0)$ $B = (0,-10)$ $AB = AC$ Using the angle between $AB$ and $AC$, how are the coordinates at C calculated?
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1answer
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How to prove the midpoint of a chord is also the midpoint of the line segment defined by the points of intersection of other two chords with it?

Bernhard Elsner, alias MathOMan, posted this exercise in plane Geometry, Theorem about a circle, three chords and a midpoint on January 29th, 2010. "Let $\mathcal{C}$ be a circle, $A,B$ two distinct ...
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509 views

Dividing a disk into 7 equal pieces with 3 line segments

Can you divide a disk into 7 pieces of equal area, with 3 line segments? (You can surely divide it into 7 pieces, but could those have equal areas?) (This question was left unanswered at another ...
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3answers
416 views

Rigorous synthetic geometry without Hilbert axiomatics

Are there books or article that develop (or sketch the main points) of Euclidean geometry without fudging the hard parts such as angle measure, but might at times use coordinates, calculus or other ...
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2answers
781 views

Distinct Hamiltonian cycles of the icosahedron and dodecahedron

I am seeking a listing of the distinct Hamiltonian cycles following the edges of the icosahedron and the dodecahedron. By distinct I mean they are not congruent by some symmetry of the icosahedron or ...
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1answer
577 views

Collective Term for $XY, YZ$ and $ZX$ Planes

Is there a collective term for the $XY, YZ$ and $ZX$ planes in $3D$ co-ordinate geometry? I was thinking "principal planes" but I'm not sure where I heard that.
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4answers
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Hyperbolic critters studying Euclidean geometry

You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise. ...
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16answers
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What is the most elegant proof of the Pythagorean theorem?

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's the most elegant proof? My favorite ...
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3answers
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Proof of Angle in a Semi-Circle is 90 degrees

There is a well known theorem often stated as the angle in a semi-circle being 90 degrees. To be more accurate, any triangle with one of its sides being a diameter and all vertices on the circle has ...
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1answer
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RHS Congruency test - What makes 90 degrees different?

RHS is a well known test for determining the congruency of triangles. It is easy enough to prove it works, simply use Pythagorus' theorem to reduce to SSS. I thought that it seems strange that this ...
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Compass-and-straightedge construction of the square root of a given line?

Given A straight line of arbitrary length The ability to construct a straight line in any direction from any starting point with the "unit length", or the length whose square root of its magnitude ...
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4answers
620 views

What transformations of the plane are geometrically constructable (compass & straight edge)?

Congruence transformations (isometries) and similarity transformations (isometries + dilations) should be constructable. What about other affine transformations? Other conformal mappings? edit: by ...
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Software for solving geometry questions

When I used to compete in Olympiad Competitions back in high school, a decent number of the easier geometry questions were solvable by what we called a geometry bash. Basically, you'd label every ...