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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4
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1answer
861 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 ...
2
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1answer
242 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
242 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 ...
0
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1answer
158 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 ...
3
votes
2answers
6k 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 ...
0
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2answers
622 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 ...
3
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0answers
93 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
91 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, ...
3
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1answer
279 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 ...
2
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1answer
302 views

geometric construction of a given angle

Given any angle how can you say that it is constructable or not?
1
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1answer
953 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 + ( ...
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votes
2answers
2k 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?
3
votes
2answers
109 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
votes
1answer
418 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? ...
23
votes
7answers
14k 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
votes
1answer
175 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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vote
1answer
115 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 ...
14
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2answers
794 views

Finding the circles passing through two points and touching a circle

Given two points and a circle, construct a/the circle through the two points and touching the given circle. I came across this problem in History of Numerical Analysis by H. Goldstein. I spent some ...
19
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6answers
2k views

Is it possible to solve any Euclidean geometry problem using a computer?

By "problem", I mean a high-school type geometry problem. If no, is there other set of axioms that allows that? If yes, are there any software that does that? I did a search, but was not able to ...
7
votes
2answers
5k views

Proof that the angle sum of a triangle is always greater than 180 degrees in elliptic geometry

I've scoured the internet and have found many proofs showing that in Euclidean geometry, the angle sum of a triangle is always 180 degrees. I've also found many proofs showing that in hyperbolic ...
4
votes
1answer
302 views

A basic question on the terminology used in Desargues' Theorem

This is a pretty elementary concerning the terminology commonly used in Desargues' Theorem from plane geometry (or really, projective geometry). At least in some representative cases, I totally buy ...
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0answers
157 views

Motivation for studying compass and straightedge constructions? [duplicate]

Possible Duplicate: What is the (mathematical) point of geometric constructions? Are there good motivations to study compass and straightedge constructions? More specifically I want to ...
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0answers
243 views

Calculating the Epsilon Neighborhood of line segments in 3d

I am working on a trajectory clustering algorithm (in C++) and one of the steps required in this algorithm is to take a set of 3d line segments (D), and for each line segment (L) in D, to calculate an ...
10
votes
1answer
344 views

Chebyshev center = center of mass?

I would like to know for which convex polyhedra $P$ in $\mathbb{R}^3$, is the center of the largest sphere enclosed in $P$ (a.k.a. the Chebyshev center, or the incenter) the same as the center of ...
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votes
3answers
1k views

Isosceles trapezoid

I was solving an exercise on Isosceles trapezoid whose diagonal was given, and I note that If I draw a diagonal in the isosceles trapezoid I got two triangles To determine the area of the triangles I ...
5
votes
2answers
883 views

Finding point on a circle

I know how to find a point on a circle given a radius and an angle, but my knowledge of trigonometry doesn't extend much further than that. My question is probably best explained diagrammatically: ...
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1answer
78 views

Length of sum of high-dimensional 3 vectors in terms of their lengths and inner products

I have 3 vectors in a high-dimensional euclidean space. I would like to express the length of their sum in terms of their lengths and inner products. For two vectors, I can do it with the law of ...
7
votes
1answer
231 views

What types of geometries are scale-invariant?

This question explains that scale-invariance (or more accurately, similarity) is an important property of Euclidean geometry. Are there any other ways to define scale-invariant geometries in any ...
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5answers
631 views

What are the postulates that can be used to derive geometry?

What are the various sets of postulates that can used to derive Euclidean geometry? It might be nice to have several different approaches together for comparison purposes and for ready reference. It ...
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votes
3answers
2k views

Why is Euclidean geometry scale-invariant?

In Euclidean geometry, I frequently use concepts related to invariance under scaling. For example, I know that if two squares have different side lengths, the ratio of their side lengths is the ...
8
votes
1answer
1k views

The Pythagorean theorem and Hilbert axioms

Can one state and prove the Pythagorean theorem using Hilbert's axioms of geometry, without any reference to arithmetic? Edit: Here is a possible motivation for this question (and in particular for ...
17
votes
6answers
867 views

Geometrical construction for Snell's law?

Snell's law from geometrical optics states that the ratio of the angles of incidence $\theta_1$ and of the angle of refraction $\theta_2$ as shown in figure1, is the same as the opposite ratio of the ...
3
votes
1answer
3k views

Getting the third point from two points on one line

My question is the following How can I get point $(x3, y3)$ from points $(x1, y1)$ and $(x2, y2)$ ? The distance of point $(x3, y3)$ from $(x1, y1)$ is $300$.
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votes
2answers
512 views

The Dido problem with an arclength constraint

It is well known that the solution to the classical Dido problem is a semicircle, and that the solution to the classical isoperimetric problem is a circle. It's also reasonably obvious that the ...
2
votes
2answers
385 views

Concentric circles and orthogonality

Let $C_1, C_2$ be two concentric circles in the extended complex plane. Is it true that if another "circle" $C$ is orthogonal to both $C_1$ and $C_2$, then $C$ must be a line? I think that this ...
3
votes
3answers
612 views

Three non-coplanar lines in the 3D-space always have a fourth one that intersect them all?

If I have three lines $a,$ $b$ and $c$ in the euclidean 3D space, which are pairwise non-coplanar, is there always a fourth line $x$, that intersects theses three lines?
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2answers
1k views

Centroid, orthocentre, incentre, circumcentre problem

How to prove that in an isosceles triangle circumcenter, centroid, orthocenter & incentre are collinear?
14
votes
1answer
570 views

What's the average width of a convex polygon?

If one computes the average width of a triangle, then one gets $(s_1+s_2+s_3)/\pi$, where $s_1$, $s_2$, $s_3$ are the side lengths. I did this by brute force, using an integral which went through an ...
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1answer
208 views

Triangle and rational numbers

A triangle has rational side lengths and rational angles measured as degrees. Is such a triangle necessary equilateral?
3
votes
3answers
2k views

maximum number of collinear points?

I know this is a very standard question widely popular in the Internet and the Mathworld. I myself have solved the above problem is N^2 Log N avoiding floating arithmetic.However, can anyone give me a ...
4
votes
3answers
154 views

Is concavity of a real-valued function on a Euclidean space implied by concavity of its restriction to every lower dimensional affine subspace?

Consider a function $f$ over $\Re^n$ to $\Re$. Suppose it is true that for every affine subspace with dimension strictly lesser than $n$ the function $f$ is concave. Is the function $f$ concave over ...
1
vote
2answers
125 views

Radius of a hypercube at a given angle

For a ray from the origin with a given angle in $R^n$, I am trying to find the radius at which that ray intersects the frontier of the unit n-cube. In two dimensions, the picture is this: Given ...
4
votes
2answers
740 views

Euclidean Version of Pappus's theorem

I'm going through Hartshorne's Geometry, and one of the exercises has stumped me for a good few hours. The problem is a version of one of Pappus's theorems: Let $A$, $B$, $C$, be points on a line ...
0
votes
2answers
716 views

Math problem- geometry- arbitrary points

an arbitrary point P is chosen on side BC of triangle ABC and perpendiculars PU and PV are drawn from P to other two sides of the triangle. (It may be that U or V lies on an extension of AB or AC and ...
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votes
3answers
1k views

In neutral geometry, can a family of parallel lines leave holes in the plane?

In neutral plane geometry, Euclidean geometry without the parallel postulate, I want to show that the family of parallel lines all perpendicular to a given line pass through all of the plane, leaving ...
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votes
3answers
940 views

Given a line segment $AB$ and a point $P$ inside a circle, can one construct a chord through $P$ congruent to $AB$?

This is a little exercise found in Robin Hartshorne's Euclid: Geometry and Beyond: I believe I have found a solution, which only exists when $AB$ has length at least that of the chord bisected by ...
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4answers
7k views

Finding the area of a quadrilateral

I have a quadrilateral whose four sides are given $2341276$, $34374833$, $18278172$, $17267343$ units. How can I find out its area? What would be the area?
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2answers
124 views

How to use model of point movement

I have got discrete time model of the point x movement to point y stated as: $$z = x + s\left( \frac{y - x}{||y-x||} \right)$$ where ...
5
votes
2answers
307 views

name of a shape

Let P be a point, not the center, in the interior of a (round) disk D⊂ℝ² and let A and B be points on ∂D such that the line segments AP and BP have equal length. Choose an arc AB. What's the shape ...
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0answers
105 views

Complexity of Counting the number of inducing $n$-gons

Definition: A $n$-gon is simple if it has no self intersection and is in general position if no pair of its edges are parallel. It is clear that by extending the edges of each simple $n$-gon in ...