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.

learn more… | top users | synonyms (1)

0
votes
1answer
10 views

Equation of a line about which we are reflecting

Let $A$ be the matrix of a reflection about a line of the euclidean plane (w.r.t. the standard basis). How can I find the equation of the line?
5
votes
1answer
63 views

Points on a 2D plane spanned by a turtle graphics system

Suppose you have a turtle graphics system with a set "turning angle" $\delta$, in which the turtle can execute three commands: $F$: Go forward, by unit length, in the current direction. (The initial ...
1
vote
3answers
23 views

Reflections in Euclidean plane

Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be the counterclockwise rotation of $\frac{\pi}{2}$ and $S: \mathbb{R}^2 \to \mathbb{R}^2$ be the reflection w.r.t. the line $x+3y=0$. There exists a reflection ...
4
votes
2answers
2k views

How to calculate volume of 3d convex hull?

Convex hull is defined by a set of planes (point on plane, plane normal). I also know the plane intersections points which form polygons on each face. How to calculate volume of convex hull?
0
votes
0answers
25 views

On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.

So, according to this figure : http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII23.html We cannot have similar segments of circles and unequal ones be built on the same side of the same ...
3
votes
2answers
702 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 ...
4
votes
2answers
228 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 ...
0
votes
2answers
1k views

Centroid, orthocentre, incentre, circumcentre problem

How to prove that in an isosceles triangle circumcenter, centroid, orthocenter & incentre are collinear?
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 $300$.
0
votes
1answer
24 views

In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base.

I have the following theorem : "In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base." (Figure is in the link) ...
2
votes
2answers
158 views

Find the Outgoing Edge with the Smallest Angles, Given one Incident Edges and Multiple Outgoing Edges

I have one incident edges and multiple outgoing Edges, for which I want to pick an outgoing edge such that the angles between the outgoing edge and the incoming edge is the smallest of all. We know ...
9
votes
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 ...
4
votes
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 ...
1
vote
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?
-1
votes
3answers
51 views

How can I find the diagonal of a quadrilateral? [closed]

Given a quadrilateral $MNPQ$ for which $MN=26$, $NP=30$, $PQ=17$, $QM=25$ and $MP=28$ how do I find the length of $NQ$?
91
votes
8answers
3k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (June. 2015) This question has been moved to MathOverflow, where a recent write-up finds a similar approximation as leonbloy's post below; see here. Randomly break a stick in five places. ...
21
votes
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)?
1
vote
0answers
27 views

Horn angles and Euclid's elements.

We have the following statement by Euclid : "I say further that the angle of the semicircle contained by the straight line BA and the circumference CHA is greater than any acute rectilinear angle, and ...
1
vote
1answer
35 views

Euclid's elements proposition 15 book 3

http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII15.html I have understood the proof in general. It is only a small detail which i'm not sure. Maybe it's because english isn't my first ...
0
votes
1answer
49 views

Online tool for making Geometric Constructions.

There was a website where it tasked you making different geometric shapes using only a compass and straightedge. I've looked for it and I can't find it or even discussion about it. What I do remember ...
6
votes
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 ...
0
votes
1answer
36 views

Euclid's elements proposition 13 book 3

"Then, since on the circumference of each of the circles ABDC and ACK two points A and C have been taken at random, the straight line joining the points falls within each circle, but it fell within ...
1
vote
1answer
73 views

Why is the Euclidian norm used to measure complex numbers?

Why is the Euclidian norm used to measure complex numbers? The complex numbers are numbers (or more precisely, pairs of numbers), and I can't see why are they essentially connected to the ...
1
vote
1answer
45 views

How many sets of four points in an MxN grid have one point contained by three other points?

Given a 3x3 grid: 1 2 3 8 9 4 7 6 5 We find 126 distinct sets of 4 points $$\binom{9}{4}$$ There are 8 cases such that when the points are connected with a line in clockwise direction, one point ...
0
votes
2answers
40 views

Rotations/Transformations with Complex Numbers/Eulers Formula

Hello, I am not entirely sure how to do this question, as I understand a rotation in the complex plane can be described by using Euler's formula, $e^{i\theta}$. Since this is an equilateral ...
0
votes
1answer
25 views

Center and angle of complex function

Does a complex function of type $f(z)=az+b $ always have a center and angle (of rotation) or only when $b=0$ since $b\neq0$ represents a translation?
3
votes
1answer
63 views

Euclidean triangle is determined by Angle, Median and radius of exterior circle

consider two triangles $\Delta(A,B,C), \Delta(A',B',C')$ in euclidean plane. I want to prove that these triangles are congruent if they are equal in the following data: they have the same angle at ...
1
vote
1answer
68 views

Find my coordinates from distance with unknown coordinates

I am trying to work out if there is a way to calculate some coordinates relative to each other simply by knowing $3$ or more distances from some unknown points. I do not have a distance matrix, I ...
1
vote
1answer
137 views

Determine Euler Angles from look, up, and cross vectors

I have a spaceship flying through a $3D$ space. The flight is determined by applying a quaternion to the look, up, and cross vectors with the following scheme (this is working perfectly): starting ...
2
votes
1answer
640 views

Intersection of Two Circles

I have two circles as: $C_1: (x-x_1)^2+(y-y_1)^2=r_1^2$ and $C_2: (x-x_2)^2+(y-y_2)^2 =r_2^2$ and these circles have non-empty intersection. In other words $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\leq ...
0
votes
2answers
59 views

Why is this point set a circle?

consider a circle in Euclidean plane $E$ and any point $A$ in the interior of the circle. Now consider all secants $s_A$ to the circle through the point $A$. The claim is now that the set of midpoints ...
1
vote
1answer
29 views

Lengths on the unit octahedron

Consider the face of the unit octahedron, defined by: $$O^2 = \{(x,y,z): |x|+|y|+|z|=1\}$$ Every point on the octahedron has between 0 and 3 positive coordinates. E.g, in $(0.2,-0.3,0.5)$, the $x$ ...
1
vote
2answers
52 views

Show that $\triangle ABK \cong \triangle ABL$. [closed]

Show that $\triangle ABK \cong \triangle ABL$ where $D$ is the circumcenter,$K$ is the orthocenter and L lies on the circle.
2
votes
2answers
34 views

Proof of folding to trisect a right angle

If first you fold a normal (letter or A4) piece of paper in half: and then you fold one corner to meet the halfway line: Then you've trisected the right angle at bottom left - but how does one ...
0
votes
0answers
24 views

Solid angle in $D$ dimensions

Consider $d\Omega_D$, the element of solid angle in dimension $D$. Suppose an integrand depends only on $m-1$ of the angles (so that it doesn't depend on the set $\left\{\phi_i, i=1,\dots,D-m\right\}$ ...
3
votes
0answers
52 views

About pythagorean triples

In the circle of diameter $AB$ it is well known each point $C$ determines a right triangle $\Delta ABC$ and so it is with every point $D$ on the circle of diameter $AC$ determining a right triangle ...
0
votes
2answers
45 views

How do I calculate the angles between a point on a sphere and each unit vector in $\Bbb R ^3$?

Given the Cartesian coordinates of any point $p$ on the surface of a sphere in $\Bbb R ^3$, how do I calculate the angles between each axis $(x, y, z)$ and the vector $n$ defined by origin $o$ and ...
0
votes
2answers
52 views

any good sourcebook for plane geometry problems?

I wanted to find some good resource books for euclidean plane geometry. would anybody name some titles?
3
votes
1answer
49 views

Surface Area of unit n-sphere covered by rotating a unit vector around a fixed unit vector such that angle between the two vectors is always fixed.

Consider an n-dimensional unit sphere and unit vector from the origin with its tip lying on the surface of sphere. Consider another vector which makes some angle say $\epsilon$ with unit vector. From ...
5
votes
2answers
104 views

What's interesting in latus rectum?

I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical ...
0
votes
1answer
40 views

Angle between edge and lateral face of a regular pyramid

MABCD is a regular pyramid (ABCD is square and the lateral edges are equal). The angle between the base ABCD and the plane through BD, which is perpendicular to MC, is $\phi$. Then what is the sine of ...
1
vote
1answer
110 views

In any triangle the angle opposite the greater side is greater.

I have a small problem with the following : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI18.html I did understand the proof, but the proposition claims that the angle opposite the ...
1
vote
0answers
23 views

Faster Alternative than Calculating Euclidian Distance to determine which Coordinate has Max Distance from a fixed coordinate (eg (0,0))

I am developing a program that needs me to determine which coordinate in a 2-d figure has maximum distance from a fixed coordinate. Let me demonstrate: 3 points: (1,3), (2,2), (5,0) ; Fixed point: ...
1
vote
0answers
8 views

Construct a matrix satisfying a linear restriction with bounded singular values

Suppose for some matrix $A\in\mathbb{R}^{n\times n}$, and some vectors $x,y\in\mathbb{R}^n$, $y=Ax$. Under what conditions (ideally on $A$) can one construct a matrix $B\in\mathbb{R}^{n\times n}$ ...
1
vote
1answer
26 views

Straightedge-Only for Perpendicularity

Given a triangle ABC and a midpoint M (of the line AB), is it possible to check whether the line CM is perpendicular to AB with a straightedge only? By this, I mean that points can be added ...
2
votes
1answer
26 views

Finding distance from point to line which is perpendicular to another line

Find the distance of the point $(1,1,1)$ from $x+y+z=1$ measured perpendicular to the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{6}$
0
votes
0answers
22 views

Is there a term for two distance metrics that give the same ordering?

In order to calculate and sort things by distance from each other in a computer program I need to find an easy and fast distance metric to calculate. I can probably find one myself. However, first I ...
4
votes
1answer
70 views

Moving circular disk between two parallel sinusoidal curves

Find the largest radius of the circle that can be "rolled" between the curves $y = sin(x)$ and $y = sin(x)+1$. After two weeks of research, I finally give up.
4
votes
2answers
95 views

Describing convex hulls in purely metrical terms

Let $X$ denote a Euclidean space; take $X = \mathbb{R}^n$ for concreteness. Now consider $x,y \in X$. Then the line segment joining $x$ and $y,$ hereafter denoted $[x,y]$, can be described in ...
0
votes
1answer
52 views

Proof that an angle across line is equal to 180 degrees

When given a straight line, how do you prove that an angle across it is equal to 180 degrees, or two right angles? It feels like something that should be an axiom, but it isn't one of the 5 ...