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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401 views

PQRS is a cyclic quadrilateral. If SQ bisect <PQR prove chord PS = chord SR

$PQRS$ is a cyclic quadrilateral. If $\overline{SQ}$ bisects $\angle PQR$ prove chord $\overline{PS}$ = chord $\overline{SR}$.
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3answers
3k views

Find the perimeter of any triangle given the three altitude lengths

The altitude lengths are 12, 15 and 20. I would like a process rather than just a single solution.
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2answers
520 views

geometric meaning of a trigonometric identity

It follows from the law of cosines that if $a,b,c$ are the lengths of the sides of a triangle with respective opposite angles $\alpha,\beta,\gamma$, then $$ a^2+b^2+c^2 = 2ab\cos\gamma + 2ac\cos\beta ...
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1answer
217 views

Maximal mapping of a convex set to the unit disk

EDIT: To make my question more precise i think we can narrow it down to this. Say you have a simple polygon that includes the origin, that is completely contained in the unit disk, we can 'blow up' ...
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3answers
629 views

Projection of tetrahedron to complex plane

It is widely known that: distinct points $a,b,c$ in the complex plane form equilateral triangle iff $ (a+b+c)^{2}=3(a^{2}+b^{2}+c^{2}). $ New to me is this fact: let $a,b,c,d$ be the images of ...
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1answer
181 views

Approximating convex sets with disjoint rectangles in an optimal way

Let $O \subset \mathbb{R}^2$ be a convex open set of finite Lebesgue measure $1=m(O)$. Let's call a collection $P$ of $n$ disjoint open rectangles contained in $O$ a "partial cover of $n$ pieces". ...
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1answer
710 views

3 reflections leads to glide

In a previous problem I asked about the notation used here.. I'm still not sure how to show it even though I now get what it's asking. The following is an exercise from The Four Pillars of Geometry. ...
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2answers
315 views

Circle bitangent angles

Say we have two circles $C_1$ and $C_2$ with radii $r_1$ and $r_2$, respectively. Let their centers be $d$ units apart. There are 4 bitangents, two outer and two inner. Examine the intersection of an ...
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2answers
233 views

Every 3D rotation must have an axis?

For a 1D object 1D space we can translate $x\mapsto x+a$ but cannot define a rotation as $x\mapsto ax$ would not leave the distance invariant between two points and hence the onyl rigid tranformation ...
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1answer
501 views

Property of an ellipse

I need proof for the following question. Also, I want to know, can we apply the same for other conics. If yes, where and when... Please explain. Show that there exists a point K on the major axis of ...
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2answers
3k views

What is a straight line?

I have researched this question for days and can not locate a good answer. It could be a mathematical object that is defined by an axiom as Euclid or Hilbert. But if a curve is drawn between two ...
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1answer
1k views

Geometry Reflection Notation

The following are exercises from The Four Pillars of Geometry; I'm not sure what they are stating, for example I don't know what the addition of prime (an apostrophe) to a line means. There are no ...
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5answers
12k views

Determining if an arbitrary point lies inside a triangle defined by three points?

Is there an algorithm available to determine if a point P lies inside a triangle ABC defined as three points A, B, and C? (The three line segments of the triangle can be determined as well as the ...
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2answers
249 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
213 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
196 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
193 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
603 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
485 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
167 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 ...
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1answer
93 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
99 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
118 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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4answers
1k 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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1answer
1k views

The Dihedral Angles of a Tetrahedron in terms of its edge lengths

I am interested in any references which discuss a general formula for the dihedral angles of a tetrahedron in terms of its six edge lengths. If there is a well known formula could someone please post ...
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6answers
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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
1k 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 ...
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1answer
196 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
584 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
325 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 ...
6
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1answer
206 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.
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1answer
411 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
180 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
304 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
667 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
137 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
286 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
65 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
747 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
475 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 ...
5
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1answer
691 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 ...
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2answers
700 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
113 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 ...
2
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1answer
239 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 ...
4
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1answer
298 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 ...
4
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3answers
529 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, ...
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0answers
89 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 ...
3
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1answer
668 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 ...
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1answer
186 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 ...
2
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3answers
215 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 ...