Questions tagged [euclidean-geometry]

Geometry assuming the parallel postulate: 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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5
votes
2answers
424 views

Locus and concurrent lines

This will be my first question :-) Let $\mathcal{D}_1$ and $\mathcal{D}_2$ two concurrent lines, and $F$ a point in the plane, and $H$ and $G$ its images by the symmetries of axis $\mathcal{D}_1$ and ...
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1answer
338 views

Triangle from lengths of angle bisectors

According to http://www.cut-the-knot.org/triangle/TriangleFromBisectors.shtml it is impossible to construct a triangle from the lengths of its angle bisectors. Is there a more comprehensive account of ...
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2answers
271 views

Distribution or bounds for maximum Cartesian coordinate sampled from the sufarce of an n-sphere

It's been said that for high dimensions a hypersphere is "nearly all equator". The amount of space near the poles is just ridiculously small. This of course means that from a uniformly random sample ...
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1answer
231 views

Projection of 5 skew lines

Given five skew lines, is it possible to find a point $P$ and a plane $\pi$ such that the projections of the five lines from $P$ onto $\pi$ intersect in the same point $Q$? [editet: rewritten clearly, ...
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4answers
1k views

Geometry/ Similar Triangles Problem

Consider the trangle shown below with vertices A, B, C where point D lies on the side AB, point E lies on the side BC and point F lies on the side AC and the three lines AE, BF, and CD intersect at a ...
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2answers
4k views

Calculate measurements for a diagonal fence beam

Given the width W and the height H of a rectangle, and the thickness T of a beam extending exactly from the upper left corner to the lower right corner as shown, how do I solve for length X and angle $...
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2answers
470 views

Finding a random vector exactly yay far from another point in 3D space

So I am trying to find a vector a certain distance away from another point ( the distance varies based on an input ) and I've figured out that distance^2=(newPoint-centerPoint):Dot(newPoint-...
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1answer
131 views

Arrangements of congruent rectangles

I have stumbled on an interesting problem. How many congruent rectangles on a plane can be arranged in such a way that they all touch each other but never overlap? I pondered this problem for a few ...
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3answers
150 views

Acute triangle: circles with diameters of two sides meet on the third

Let an acute triangle ABC be given. Prove that the circles whose diameters are AB and AC have a point of intersection on BC. How do I go about this problem? Can You Please Give Me a Hint?
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2answers
191 views

Calculating max points able to be placed inside a hypercube

Given a hypercube of width W, how can I figure out the maximum number of points I can place inside the hypercube such that each point is at least equal to or further than euclidean distance X away ...
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5answers
6k views

Area Between Three Circles of Differing Radii

From the link in wikipedia http://web.gnowledge.org/wiki/index.php/Area_Between_Three_Circles_of_Differing_Radii OPEN QUESTION: What is the equation, in three variables, relating the radii of ...
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6answers
7k views

Product of slopes is -1 iff perpendicular proof from first principles

Once again I'm working through Stillwell's Four Pillars of Geometry. I'm on Chapter 3 where he first introduces coordinates. The question reads, 3.5.1 Show that lines of slopes $t_1$ and $t_2$ ...
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1answer
2k views

Solid body rotation around 2-axes

I am trying to understand how to describe the rotation of a solid body flying in 3D space. From physics forums, I understand that the rotation of any solid object in space, is around 2 axes ...
3
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1answer
855 views

Probabilities of Non-Regular Dice

Thinking about dice: for all the Platonic solids, it's very easy to figure out the odds of a particular face landing face-up in a roll of the die. If I have an arbitrary 6-sided solid, how do you ...
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2answers
2k views

Expanding (or reducing) a shape

If I want to expand or reduce a shape what mathematical methods are there to do this. I'd like to understand scaling which seems simple enough. Using my limited knowledge I would do this by measuring ...
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4answers
512 views

Term for Tetrahedron with Three Right Angles at a Point

Is there a name for the tetrahedron/pyramid (four vertices, four triangular faces, six edges) where three edges meet orthogonally at a point? Three of the faces are right triangles. Another ...
7
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1answer
408 views

Efficient algorithm for finding how many times a point is inside the triangles formed by given points

Given n 2D points and a special point p, what would be the best way to find how many times p is inside among those $^nC_3$ triangles formed by the n points.
15
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1answer
612 views

Two tetrahedra are congruent given a certain condition

This question is inspired by a Miklos Schweitzer problem, namely Problem 9./2007 Let $A$ and $B$ be two triangles in the plane such that the interior of both triangles contains the origin, and for ...
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0answers
268 views

Distance between two unbounded sets

How to calculate the distance between two (possibly unbounded) ranges of positive real numbers? For example, if three guys specify their prices they would pay for a product: ...
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1answer
1k views

I want to rotate a point on a sphere surface

I want to rotate a point on a sphere surface . I was instructed as I can use Rodrigues rotation formula , (I thank ja72 very much). I tried to use the formula but it did not work . I can not find ...
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1answer
1k views

Arc length of a great circle which is the hypotenuse of an isoceles right triangle on the sphere

I am doing a problem which requires me to find the arclength of the hypotenuse of an isosceles right triangle. (The book calls it a 2 Dimensional Sphere but I hope that is a typo) I start at the ...
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3answers
154 views

Get Point Cloud from Complete Weighted Graph

Is it possible to calculate the x,y position of a node in a complete weighted graph with euklidic metric when i have the weight of every edge in the graph? it would be really usefull for me to plot ...
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1answer
536 views

The formula for the point position rotated around an axis of a sphere

Please let me know the formula for the point position rotated around an axis of a sphere. In detail, I want to do as follows. Given: any point $p_1$ to decide the rotation axis ax of a sphere of ( ...
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1answer
71 views

How do I find a point $(x_1,y_1)$ if I have an origin point $(x_0,y_0)$, a distance, and $\theta$?

I'm trying to figure this out for player movement in a video game but I'm having trouble figuring it out: How do I find a point $(x_1,y_1)$ if I have an origin point $(x_0,y_0)$, a distance, and $\...
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2answers
2k views

Aren't asteroids contradicting Euler's rotation theorem?

I am totally confused about Euler's rotation theorem. Normally I would think that an asteroid could rotate around two axes simultaneously. But Euler's rotation theorem states that: In geometry, ...
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2answers
638 views

Diffraction and Computer Generated Holography Calculations

I've tried this through Mathematica, and hit my own limit in math ability trying to do this, both to no avail. I'm assuming there is no way to do so, as a simple solution to this problem would be a ...
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1answer
2k views

Prove three sides make a triangle from basic assumptions

I've been working through The Four Pillars of Geometry by John Stillwell. In exercise 2.5.3 he asks, How can we be sure that lengths $a,b,c>0$ with $a^2+b^2=c^2$ actually fit together to make a ...
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3answers
1k views

If the Minkowski sum of two convex closed sets is a Euclidean ball, then can the two sets be anything other than Euclidean balls?

If for two convex closed sets $S_1$ and $S_2$, the Minkowski sum is a Euclidean ball then can $S_1$ and $S_2$ be anything other than Euclidean balls themselves. I suspect they can be but I haven't ...
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1answer
131 views

equation of the hyperplane orthogonal to the general line v

Given a line v in $R^n$ from point a to point b, what is the general equation of the hyperplane that passes through a and is orthogonalto v? Ideally I am looking for the general solution in arbitrary ...
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1answer
397 views

Existence of $\pi$ [duplicate]

Possible Duplicates: Why is the ratio of the circumference of a circle to its diameter independent of the circle? Proof that Pi is constant (the same for all circles), without using limits ...
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1answer
643 views

Similar - perspective triangles implies corresponding sides are parallel?

In a general homothetic transformation, if two triangles have corresponding sides parallel then the lines joining respective vertices are concurrent at the homothetic center. I was wondering if the ...
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3answers
998 views

Dissecting a square into congruent pieces that all touch the centre

Edited for clarity: I thought I had a complete set of solutions to this: Cut a square into identical pieces so that they all touch the center point. It became clear, after some discussions, that ...
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0answers
108 views

Solving some geometry problems using involutions

Some geometry problems ( like this and this ) have short solutions if we use involutions. What references are there for solving geometry problems using involutions? I am particularly interested in ...
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1answer
2k views

Reflecting a point over a line created by two other points

This problem came up while discussing using a simplex to solve systems of equations. (By the way, yes, this is very similar to this one.) Given three points, how do I find the location of the point ...
3
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1answer
2k views

Distance between two ranges

I'm working on a clustering algorithm to group similar objects that are represented by ranges of real numbers. Let's say that I have a group of people who are buying sugar. Each of them defines ...
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3answers
2k views

Sum of coefficients of an orthogonal matrix

Let $(a_{ij})_{1 \le i,j \le n}$ be a real orthogonal matrix. Show that $$\left| \sum_{1 \le i,j \le n} a_{ij}\right| \le n.$$ Naively applying the Cauchy-Schwarz inequality only gives $n^{\frac{3}{...
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1answer
179 views

Intersecting a polygon with four points

Assuming you have four points in general position in the plane and a (possibly non-convex) polygon. How do you find the parameters of a transformation [s*R, t] (homogenous scaling, rotation and ...
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2answers
12k views

Prove converse Thales theorem, proportional sides imply parallel lines

I'm going through John Stillwell's Four Pillar's of Geometry and trying to follow the book's structure when doing the exercises. Generally, a 'pillar' is divided into two chapters; the first chapter ...
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2answers
1k views

In △ ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees?

In △ABC, D is the midpoint of AB, while E lies on BC satisfying BE = 2EC. If m∠ADC=m∠BAE, what is the measure of ∠BAC in degrees? I know already that angle A and angle D are congruent because m∠ADC=m∠...
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0answers
100 views

Computing the proportion of vectors with the same sign

Let $s \propto (1, \ldots, 1)$ be a d-dimensional unit vector, so that $s_i = 1 / \sqrt{d}$. Let ${\cal V} = \{ u \in {\mathbb S}^{d-1}:u \cdot s = \cos \theta \}$ be a collection of unit vectors that ...
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1answer
102 views

When is the hexagon formed by the feet of perpendiculars and the midpoints of sides regular?

In a $\Delta ABC$, consider the hexagon $m_{a}f_{a}m_{b}f_{b}m_{c}f_{c}$ where $m$ and $f$ stand for the midpoint and foot of perpendicular of the respective sides. Now it can be shown quite easily, ...
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1answer
1k views

A question related to Plane and Sphere

The problem: A variable plane passes through a fixed point (a,b,c) and cuts the coordinate axes at P, Q, R (where none of P, Q, R is the origin). The co-ordinates (x,y,z) of the center of the sphere ...
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1answer
356 views

geometric construction of a given angle

Given any angle how can you say that it is constructable or not?
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1answer
2k views

Formula for the coordinate of the midpoint in spherical coordinate system

Please let me know the formula for the coordinate of the midpoint of 2 points in spherical coordinate system . If possible , I want the answer includes the exact formula as , midpoint = point1 + ( ...
5
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2answers
8k views

two point line form in 3d

the two-point form for a line in 2d is $$y-y_1 = \left(\frac{y_2-y_1}{x_2-x_1}\right)(x-x_1);$$ what is it for 3d lines/planes?
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2answers
127 views

Is this lemma about the minimal distance of two lines true?

In school, I recently proved a solid geometry excercise by assuming that the following lemma is true: If two lines $g$ and $h$ in the euclidian space are not parallel, and if the lines seem ...
2
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1answer
438 views

3d axis rotation

I have a vector V= and several line segments Seg1, Seg2, Seg3, Seg4. I want to know how to rotate each of the line segments so that the X axis is parallel to my given vector. How can I do this? ...
39
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9answers
35k views

Book recommendation on plane Euclidean geometry

I consider myself relatively good at math, though I don't know it at a high level (yet). One of my problems is that I'm not very comfortable with geometry, unlike algebra, or to restate, I'm much more ...
5
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1answer
197 views

euclidean geometric construction

this is question 42 in the red book of mathematical problems by k. s. williams and k. hardy. let abcd be a convex quadrilateral. let p be the point outside abcd such that $|ap| = |pb|$ and $\angle ...
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1answer
132 views

Converting an angle in Euclid proof-wise into a relation on 'Cartesian' polynomials

In Is it possible to solve any Euclidean geometry problem using a computer? I claimed that one can convert the statement of a theorem in Euclid into multivariate polynomials such that Groebner basis ...