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
3answers
97 views

Perpendicular lines inside and outside a circle

No trigonometry allowed. Let $\Delta ABC$ be inscribed inside a circle.Let $P$ be a point on the circle.Let $PD$ and $PE$ be perpendiculars on on $BC$ and $AC$ respectively.Let $DE$ when extended ...
0
votes
1answer
84 views

Pitch, roll, yaw rotations?

I have a random orientation in a room described by pitch, yaw, and roll angles. When I do ...
5
votes
2answers
2k views

Equation of Earth's Orbit around Sun (ellipse)

The preihelion is the smallest distance from a planet to the sun, and aphelion is the greatest distance. The sun is one of the two foci. For the Earth, the perihelion is 147.1 million km and the ...
0
votes
3answers
89 views

Equation of a line passing through a given point, perpendicular with a line [closed]

I need the equation of the line passing through a given point $A(2,3,1)$ perpendicular to the given line $$ \frac{x+1}{2}= \frac{y}{-1} = \frac{z-2}{3}. $$ I think there must bee some kind of rule ...
2
votes
1answer
70 views

Help on whether a geometry solution is valid.

Let $ABC$ and $AB'C'$ be similar right angled triangles with right angles at $C$ and $C'$, respectively. Let $l$ be the line between $C$ and $C'$, and let $D$ and $D'$ be the points on $l$ such that ...
-1
votes
1answer
174 views

Spherical coordinates to cartesian coordinates.

I want to find out the distance between the centers of $2$ circles. Say, circle $1$ $(\theta,\phi)$ circle $2$ $(\theta,\phi)$ The radius of this circle is found using $d\tan(\theta)$ where $d$ is ...
8
votes
2answers
122 views

The angle $168^\circ$ is constructible

Prove that the angle $\theta=168^\circ$ is constructible using a straightedge and a compass. It is enough to show that the number $\cos\theta$ is constructible, and WolframAlpha gave ...
1
vote
1answer
48 views

Heptagonal tesselations

Are there any tesselations of the Euclidean plane that use only regular polygons such that one of them is a heptagon? If so, what is the tesselation that uses the fewest different types of polygon ...
0
votes
1answer
27 views

Calculating incremental coordinate change along a 3D vector

This is pretty trivial, but I've not managed to resolve it to my satisfaction and it has reached the stage where fresh eyes are definitely useful! I have an xyz point, and a 3D vector originating at ...
0
votes
4answers
132 views

Maximizing the Magnitude of the Resultant Vector

Given a set of $n$ two-dimensional unit vectors: $\left\{ \mathbf{v}_1, \dots, \mathbf{v}_n \right\}$, I want to find the coefficients $\left\{ \alpha_1, \dots, \alpha_n \right\}$, $0 \leq \alpha_i ...
0
votes
1answer
65 views

What class am I most prepared for?

I've only taken up to calc 3, discrete, and linear algebra. Which course am I most prepared for? I'm going to be taking differential equations and advanced calc, but I want to take a 3rd class. I can ...
1
vote
1answer
35 views

When two triangles have the same orthocenter and circumscribing circle, are nine points are the same too?

When two triangles have the same orthocenter and circumscribing circle, are the nine points are the same too? If two triangles have the same circumscribing circle, at least the sides have the same ...
6
votes
4answers
738 views

which axiom(s) are behind the Pythagorean Theorem

There are many elementary proofs for the Pythagorean Theorem, but no matter they use areas, similarities, even algebraic proofs, it is not straightforward to tell why it is true tracing back to the ...
0
votes
2answers
54 views

Prove $\sin^2 A = \sin^2 B \sin^2 C - 2\sin B \sin C \cos A$

I am asking for help with this proof: Given $\triangle ABC$. Prove that $\sin^2 A = \sin^2 B+ \sin^2 C - 2\sin B \sin C \cos A$
0
votes
2answers
66 views

Is there a Taylor series for vector cross product?

I have this equation, where $u,v,w,a,b,Ɵ$ are constants. The RHS comes from the Geometric definition of the LHS $(u,v,w)(a,b,c)=||(u,v,w)||||(a,b,c)||\cos(\theta)$ Expanding the 2-norms ...
2
votes
0answers
48 views

Geometric conditions equivalent to a set being the unit circle for some norm

Here's the question, as in the textbook (Real Mathematical Analysis, Pugh). The unit ball with respect to a norm $||\, \cdot \,||$ on $\mathbb{R}^2$ is $$ \{ v \in \mathbb{R}^2 : ||\, v \,|| ...
3
votes
2answers
1k 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
2answers
89 views

Solve the following problem…

My problem is: In a circle of radius $R$ is inscribed an equilateral triangle $ABC$. Through the point $C$ is drawn a line which intersects $AB$ in point $M$ and the circle, for the second time, in ...
41
votes
15answers
10k views

What is the most elegant proof of the Pythagorean theorem?

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 ...
4
votes
0answers
293 views

Counting simple quadrilaterals in a rectangular lattice.

I've been trying to make an algorithm to find the number of all possible simple quadrilaterals in a N*M lattice. I already have a brute force solution but since this is a Project Euler problem I ...
0
votes
0answers
58 views

How to prove that P, F, and E are collinear from this following parallelogram problem?

Inside parallelogram $ABCD$ with $\angle A=90^\circ$, a circle with diameter $AC$ intersects $CB$ and $CD$ at $E$ and $F$ respectively. Tangent line of this circle at $A$ intersects $BD$ at point ...
0
votes
2answers
40 views

Triangles with common centroid

Consider the points $A',B',C'$ on the sides $BC,CA,AB$ of a triangle $ABC$ respectively, such that $BA'/A'C=CB'/B'A=AC'/C'B$. Show that the triangles $ABC$ and $A'B'C'$ share a common centroid.
0
votes
2answers
46 views

how to find three vertices of a triangle.

Where s=circumcenter, H= orthocenter, and A'= midpoint of one side of triangle. How can can I determine the location of the three vertices of the triangle?
1
vote
2answers
107 views

Euclidean geometry prerequisites

I have used enrolled in a introduction to Euclidean geometry course, but I have very little experience with geometry, almost none. I have an engineering background so I have taken calculus, linear ...
-1
votes
1answer
23 views

Perimeter of the triangle

$BO$ and $CO$ are angle bisectors of triangle $ABC$. How much is perimeter of the triangle $AMN$?
0
votes
0answers
35 views

Partition of the 3d space with circles?

Does it exist a partition of the 3d space with circles of positive radium? I know the answer is no for a plane, but I can not transpose my demonstration to the space and I have no clue on how to do ...
1
vote
0answers
130 views

Find features in a Signed Distance Field

I'm currently trying to improve a meshing algorithm for signed distance fields (which are of the form $f(x,y,z) = w$, where $x,y,z$ is the location of my query, $w$ indicates the distance to the ...
0
votes
1answer
54 views

How long is the diagonal of this trapezoid?

Given a trapezoid $abcd$, with $|ab| = 1$, and angles $\angle dab = 3\theta/4$, $\angle abc = (\pi + \theta)/2$, $\angle bcd = (\pi - \theta)/2$, and $\angle cda = \pi - 3\theta/4$ (see figure below), ...
1
vote
2answers
73 views

Geometry problem about angles and triangles

I've been working on this problem for a while. It doesn't seem to hard, but I cannot reach a satisfying solution. The triangle $ABC$ is isosceles with base $\overline{AC}$. A point $O$ is also ...
1
vote
1answer
257 views

Equivalence between algebraic statements and geometric relations.

I'm currently trying to read a geometry and symmetry book and came across a little problem that I am having difficulty understanding. I need to show that if xm=mx then the point X is on the line M, ...
2
votes
1answer
87 views

Notation and hierarchy of cartesian spaces, euclidean spaces, riemannian spaces and manifolds

I am confused by some definitions. Forgive the looseness of my language. A Cartesian space is basically a space of points that can be represented by n-tuples ( and other things, but I won't go into ...
0
votes
0answers
21 views

How to prove that the definition of saddle surface is affine invariant?

I have a smooth saddle surface in $\mathbb{R}^n$, so for any normal vector the second fundamental form of the surface has $\det \leq 0$. How can I proove that the surface is still saddle if I stretch ...
1
vote
1answer
91 views

Proving the inequality of Cauchy-Schwarz in an Euclidean space. [duplicate]

It says let (G, <.,.>) be an euclidean space. Show that for all x, y belonging to G: modulus<x,y> <= sqrt<x,x> * sqrt<y,y> and in the ...
1
vote
1answer
41 views

the “unit speed” anlogue of the evolute of the curve

Given a curve, $\gamma: \mathbb{R} \to \mathbb{R}^2$ define the flow in the normal direction by $\gamma(t) + \epsilon \, \mathbf{n}(t)$. This is different from the evolute which moves at speed ...
5
votes
1answer
95 views

Rational distance from an equilateral triangle

Is there a nice proof for the following fact? In a plane, there does not exist a square such that its vertices are at a rational distance from each vertex of some equilateral triangle. What if ...
1
vote
1answer
83 views

To draw a straight line tangent to two given ellipses

How can I draw a a straight line that touches two ellipses? There are, like for two circles, 4 different solutions. I´m not interested in the analytical solution, but in the geometrical drawing, ...
3
votes
2answers
71 views

Is a quadrilateral with one pair of opposite angles congruent and the other pair noncongruent necessarily a kite?

If convex quadrilateral ABCD has congruent angles A and C and the other pair of angles B and D are not congruent, is ABCD necessarily a kite (two pairs of consecutive congruent sides but opposite ...
0
votes
1answer
190 views

proving parallel projection is onto

Background information- We are given two lines L and M and point p on L. We set up a correspondence from p<==>p' between the points of line L and M requiring segment PP'|| n for all p on line L. ...
87
votes
8answers
3k views

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

Edit (March. 2014) This question has been moved to mathoverflow; see here. Randomly break a stick in five places. Question: What is the probability that the resulting six pieces can form a ...
1
vote
1answer
62 views

Sylvester-Gallai Theorem

How is this theorem used in applications? I've been searching for it on the web but can't seem to find. Only to "correct codes". Can someone please give a few simple examples? /lost student
1
vote
2answers
51 views

Connecting square vertexes with minimal road

I have four cities in $A=(0,0),B=(1,0),C=(1,1),D=(0,1)$. I am asked to build the shortest motorway to connect the cities. How can I do that? I was thinking that first I need some compactness argument ...
0
votes
1answer
106 views

Are Euclidean distances a monotone function of inner products?

Does the sum of all pairs of inner-products of k vectors (real) have to decrease if the sums of Euclidean distances between all pairs of $k$ vectors happens to decrease? Similarly-if decrease is ...
0
votes
2answers
27 views

Orientation of vector relative to other vector

Given two directional vectors in 2D space, $\vec v=(v_x, v_y)$ and $\vec w=(w_x, w_y)$, what is the easiest way to calculate if $\vec w$ is orientated clockwise or counterclockwise relative to $\vec ...
1
vote
1answer
41 views

Tangents to a circle

For this construction, how would you show that the perimeter of the triangle $CDF$ is equal to $2BC$? Please include steps and whatnot.
0
votes
1answer
53 views

Nature of Points and Lines in Euclidean Geometry

It may be true that very few middle school student can grasp the meaning of lines and points in Euclidean geometry prior to a direct instruction. For example, it's possible that such a conversation ...
1
vote
1answer
48 views

volume of the solid

Using geometry, calculate the volume of the solid under $z = \sqrt{49- x^2- y^2}$ and over the circular disk $x^2+ y^2\leq49$. I am really confused for finding the limits of integration. Any help?
5
votes
0answers
266 views

Intuition - The Shortest Curve Between Two Points is a Line

The references below aver that the following is not crudely trivial: The shortest curve between two points is a (straight) line. An elementary school teacher construed it as follows ...
5
votes
2answers
191 views

Vertex-transitive polytope with large facet

Consider a vertex-transitive convex polytope with a facet containing more than the half of all vertices. Does it already have to be a simplex or are there other examples? I am particularly interested ...
4
votes
3answers
94 views

Proof that a line cuts in half the area of a parallelogram iff it goes through the intersection of the diagonals?

I read a theorem in a book which says that a line bisects a parallelogram iff it goes through the intersection of the diagonals. The edge case of this is of course if the line is one of the diagonal ...
9
votes
3answers
257 views

Rationale for a convention: Why use the semiperimeter in Heron's formula?

Heron's formula says that the area of a triangle whose sides have lengths $a, b, c$ is $\sqrt{s(s-a)(s-b)(s-c)}$ where $s=(a+b+c)/2$ is the semiperimeter. It can also be stated by saying that the ...