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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2answers
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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
1k 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
209 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 ...
21
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8answers
4k views

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
594 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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2answers
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
449 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
2k views

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
393 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
584 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
846 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
548 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 ...
8
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2answers
738 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
7k views

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
1k views

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 ...
11
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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
1k views

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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4answers
523 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
417 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
796 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 ...
4
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1answer
583 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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17answers
24k views

What is the most elegant proof of the Pythagorean theorem? [closed]

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
16k views

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 ...
6
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1answer
2k views

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 ...
10
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3answers
18k views

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 ...
5
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4answers
631 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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2answers
986 views

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 ...