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)

3
votes
2answers
227 views

What is the history of the use of the term “scalene triangle”?

A "scalene triangle" is a triangle with three unequal sides. As far as I can tell, this term is not in much use in serious mathematics — in fact, before I became a high school math teacher, I'd ...
2
votes
0answers
22 views

Theorem about two quadrilaterals with parallel edges

I'm looking for a name for the following theorem: If $abAB$ lie on one line and $cdCD$ lie on another line, and furthermore $ac\Vert AC,ad\Vert AD,bc\Vert BC$, then $bd\Vert BD$. One can ...
3
votes
0answers
78 views

Heron's Formula; an Intuitive or Visual Proof

I've found several proofs for Heron's formula for the area of a triangle in term of its sides, but none of them is simple and intuitive enough to show WHY the formula works. Do you know an intuitive ...
2
votes
0answers
37 views

How do we understand and visualize the meridians of a 3-sphere?

The equator of a n-sphere is (n-1)-sphere. Then the equator of a 3-sphere is a 2-sphere. Does this mean the meridian of a 3-sphere is a hemisphere (half 2-sphere)? In contrast, Wikipedia describes the ...
-3
votes
3answers
77 views

Maths question double or nothing.

An office has dimensions of $95\text{m} \times54\text{m}$. It has a sloped ceiling. one wall has a height of $34\text{m}$ and the opposite wall to a height of $30\text{m}$. Calculate the volume and ...
2
votes
1answer
30 views

is there a higher dimensional analogue of the first isogonic center?

I'm curious to know if, given four points $a, b, c, d$, you can always find a point $p$ such that last lines $pa, pb, pc, pd$ form equal angles pairwise. I'd also appreciate resources on 3d geometry ...
0
votes
1answer
64 views

abs(x)cos(x) in Fourier space

I am working on some problems concerning Fourier Transform and I am facing something I don't understand. I am trying to understand what is the representation of the function f(x)=abs(x)cos(x) in the ...
2
votes
2answers
28 views

Slicing up geometry to create triangles

When rendering object onto a screen, one must often cut up their objects into quads and triangles to allow the computer to process them and finally draw them onto a screen. I am trying to slice up a ...
0
votes
0answers
29 views

determine in what grid rhombus is a point

i have a rhombus ( i.e. diamond) grid determined by these equations ...
0
votes
0answers
29 views

A question of convergence

Let $T_1=[P_1,P_2,P_3]$ be a triangle of area one. Choose a point $P_4$ inside or on $T_1$ such that the area of triangle $T_2=[P_2,P_3,P_4]$ is half the area of $T_1$. Repeat this process in the ...
1
vote
0answers
33 views

Euclidean distance

I am studying a paper related to protein structure alignment where I find a equation to find the similarity of two nodes of a protein using a similarity measure $D$. This is given by $D(d1,d2) = 0.1 ...
2
votes
1answer
38 views

3 circles and 3 squares all inscirbed into a right angled triangle problem

This is quite a tricky question for me, but this is how far I got: My drawing may not be precise, but I do know the points of tangency. I am a little stuck now, and I would appreciate a great hint ...
4
votes
2answers
88 views

an olympiand geometrical problem

Let $ABCD$ be a convex quadrilateral and $E$ be a point on $CD$. Suppose $AC$ meets $BE$ at $F$, $BD$ meets $AE$ at $G$, and $AC$ meets $BD$ at $H$. If $FG//CD$, and the areas of $\triangle ...
0
votes
1answer
19 views

Find vector resultant in rhombus

Uhm I can't find a solution for this problem, perhaps someone can help me with a hint or a solution, thanks in advance :) $$DG=GH=HI=IG\\and\\ AE=EF=FB$$ Find resultant for U+V+W
1
vote
0answers
46 views

The Least Area For a Needle to Pass Through a Curve?

I don't know if this question is a famous one. One of my fellows asked me these questions to tease me, but I was able to find a solution for only one of these: There is one needle of length $2$ ...
1
vote
4answers
38 views

prependicular Vs prependicular bisector

We have $AH=HB$ and $BG=GC$ in the image below. Why is $AD=2\times FG$?
1
vote
0answers
53 views

Staircase Lemma

Let $S$ be a staircase-shape contained in the north-eastern quarter-plane. Let $k$ be the number of its south-western corners. In the staircase shown below, there are $k=4$ corners: In each corner ...
-1
votes
2answers
39 views

Is it possible for Euclid's theorem to generate 3 4 5?

I am writing an algorithm to generate pythagorean triplets and I was going to use Euclid's theorem, however I have been unable to make it generate the first pythagorean triplet namely (3,4,5). Is this ...
1
vote
1answer
24 views

How to compute an angle in specific counter-clockwise direction between vectors

I have one incoming vector and multiple outgoing vectors in 2D. I need compute an angle in this way: Imagine an incoming vector parallel to the x-axis. Then the angle-value "starts" below the ...
1
vote
0answers
26 views

To construct a right triangle given the hypotenuse and sum of two legs [duplicate]

NOTE: I want a hint only. A compass and a straightedge construction:Given a hypotenuse and the sum of lengths of the legs,we need to construct a right triangle. MY TRY: From any ray $BE$, ,let ...
0
votes
1answer
51 views

Draw a picture of a cube with edges a+b, and show it cut by planes that divide each edge into a segment of length a and a segment of length b.

I am reading through 4 pillars of geometry and I need some help with this question. Draw a picture of a cube with edges a+b, and show it cut by planes (parallel to its faces) that divide each edge ...
0
votes
0answers
12 views

Updating vector components based on change of starting position

I have an xyz point, and a 3D vector originating at that point. I would like to be able to shift the starting xyz point and update the 3D vector accordingly. For example: Starting at the xyz point ...
1
vote
2answers
42 views

Maximize the inradius given the base and the area of the triangle

BdMO 2013 Secondary: A triangle has base of length 8 and area 12. What is the radius of the largest circle that can be inscribed in this triangle? Let $A,r,s$ denote the area,inradius and ...
0
votes
3answers
85 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 ...
0
votes
3answers
91 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 ...
1
vote
1answer
44 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
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 ...
6
votes
4answers
728 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
52 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
64 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 ...
0
votes
4answers
126 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 ...
2
votes
0answers
47 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 \,|| ...
0
votes
2answers
86 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 ...
0
votes
0answers
52 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 ...
1
vote
1answer
32 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 ...
0
votes
2answers
35 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
45 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?
6
votes
2answers
221 views

About the hyperplane conjecture.

I have recently heard about the hyperplane conjecture and I would like to understand better the problematic behind this conjecture. The hyperplane conjecture: There exists a universal constant ...
2
votes
1answer
47 views

Solution for the value of an angle of a triangle ABC

Find value of angle m< DBC Where $$BD=DC=AC$$ $$2(m\langle BAC)=14(m\langle ABD)=7(m\langle BCD)$$ I tried hard but im out of ideas now, I know the answer is 20 but I want to know how, thanks ...
-1
votes
1answer
21 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
127 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), ...
2
votes
1answer
78 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 ...
1
vote
1answer
79 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 ...
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 ...
3
votes
2answers
70 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 ...
1
vote
1answer
40 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 ...
0
votes
0answers
25 views

Distance measures for binary data

I was wondering what are some good distance measures for binary data that have the following properties. I know that there are measures like the Jaccard index and the Dice Index, but they don't ...