For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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2
votes
1answer
44 views

Integration over ellipse

$A=\{(x,y)\in \Bbb R^2\mid \frac{x^2}{a^2}+\frac {y^2}{b^2}=1\}$. Find $\int_A (\cos x)y\,dx+(x+\sin x)\,dy$. Can someone please please give a methodological answer? Thanks a lot!
0
votes
2answers
79 views

Why $\int _c^df^{-1}\left(y\right)\:dy+\int _a^b\:f\left(x\right)dx=b\cdot d-a\cdot c$? [duplicate]

Why $\int _c^df^{-1}\left(y\right)\:dy+\int _a^b\:f\left(x\right)dx=b\cdot d-a\cdot c$ ? where f is an bijective function and $f(a)=b,f(c)=d,$ I don't understand graph... I can't see on graph this ...
1
vote
2answers
123 views

Plane intersecting all the lines

This might sound a bit stupid or ill thought, but I am having trouble visualizing it and proving it. Given a finite set $L$ of straight lines in $\mathbb{R^3}$ is it always possible to find a plane ...
0
votes
0answers
22 views

Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$?

Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$ ? Does anything change if we replace $SL$ with $GL$?
9
votes
0answers
74 views

Multiplicity of point as a zero of polynomial.

Let $C \subset \mathbb{P}_2$ be a projective curve defined by a homogeneous polynomial $P(x, y, z)$, and let $L \subset \mathbb{P}_2$ be the line $\{z = 0\}$. Assume $L$ is not a component of $C$. If ...
1
vote
0answers
27 views

Lie bracket question

I am wondering if this is correct. Suppose $X$ and $Y$ are two smooth vector fields which vanish at $p$: $X(p) = Y(p) = 0$. Also assume that $[X, Y](p) = 0$. Is it true that the derivative of the ...
0
votes
1answer
25 views

finding the value of a node in Pascal’s (a.k.a Yanghui's) triangle [closed]

Image the Pascal Triangle is on an x-y cartesian plane. so that the values of the nodes, by location are ...
2
votes
2answers
42 views

How to calculate the solid angle of a spherical rectangle from astronomical angles

Say I have 2 astronomical angle pairs defining a confined region on the visible hemisphere: (minAzimuth, minElevation) & (maxAzimuth, maxElevation) How can we calculate the solid angle of the ...
2
votes
0answers
36 views
+50

Laguerre's theorem on power of a point w.r.t. an algebraic curve

So on Wikipedia article for a power of a point there is a short section about Laguerre's theorem. The problem is, the article has no references, and whenever I'm trying to Google it the only things I ...
1
vote
0answers
12 views

spheres are not simpletic?

Reading some books on diferential geometry, a found that S^2n (with n>1) are not simpletic manifolds. They say it's because the de Rham cohomology of this spheres are R, but I do not understand this ...
0
votes
1answer
22 views

Find length of side of a triangle.

Let $ABC$ be a right angled triangle with $BC = 3, AC = 4$. Let $D$ be a point in the hypotenuse $AB$ such that $\angle{BCD} = 30^\circ$. Find the length of $CD$. I found $AB = 5$. How do we find ...
5
votes
1answer
34 views

locus of moving circles with changing radius

Suppose I a curve, $\gamma(t) = (x(t),y(t),R(t))$, which describes the centroids, $(x(t),y(t))$ and radii, $R(t)$, of an infinite number of circles parameterised by $t\in(a,b)$. I would like to find ...
1
vote
0answers
25 views

Is it possible to apply homothety to this problem

Given an arbitrary triangle ABC find points M and N on the sides AC and BC such that: $AM=MN=BN$. Is there a way to apply homothety to solve it?
1
vote
0answers
28 views

Vector Relations in Minkowski Space

Consider $\mathbb{R}^4$ equipped with the Lorentz inner product: $$\eta(X,Y)=x^0y^0-x^1y^1-x^2y^2-x^3y^3$$ Let $X,Y\in\mathbb{R}^4$, $X\not=0$ and $Y\not=0$, two future-causal (this means: ...
1
vote
0answers
43 views

Why $\pi$ is not Constructible with Circumference Length

If we use a compass to draw a circle with a diameter of length 1, then the circumference is $\pi$. From the definition given here (http://en.wikipedia.org/wiki/Constructible_number), it seems to me ...
0
votes
1answer
32 views

Existence of the Square in “Squaring the Circle” Problem

I understand that a square with area $\pi$ cannot be constructed using straightedge and compass. But such a square surely exists (and can be constructed through other means), right? If I'm right, I'm ...
0
votes
0answers
26 views

Assignment question - extended answer: radius, distances, percentage of area

Currently finishing my assignment and just want to be sure I am on the right track. This is my final extended answer question. During a full solar eclipse the moon almost exactly obscures the sun, ...
0
votes
0answers
32 views

Formula for area of circle made up of squares

I need to draw an approximate circle on a grid of squares and find its area. Each square must either be completely part of the circle or not at all occupied. Obviously, this means that it cannot be a ...
0
votes
0answers
15 views

Approximation of length of monotone curve

Recently, I am interesting about some geometry problem. One of them is: Let $f$ is monotone and has continuous derivative on $[a,b]$. Then $lf\le\Delta f+\Delta h$ where $lf$ denote the ...
0
votes
0answers
19 views

Can you partition a rectangle into exactly 3 congruent non-rectangular parts?

Recently I came upon the following result: Theorem (*): Let $n$ be a positive integer not equal to $1,3,5,7,9$. Then it is possible to partition a rectangle into exactly $n$ congruent non-rectangular ...
0
votes
1answer
33 views

Geometry problem; right triangles in a square

Given square ABCD with E the midpoint of CD. Join A to E and drop a perpendicular from B to AE at F. Assign coordinates D(0,0), C(20,0), B(20,20) and A(0,20). Find the coordinates of F. If ...
0
votes
0answers
12 views

Source sound position mutiple point

i want find sound source position like this picture : ![find source][1] But i just konw the delay I know the delay for position 2 and 3 (or more) from source after hits the first point. I don't ...
1
vote
1answer
13 views

Finding the euclidean centers of the geodesics AB, AC, and BC

I am trying to learn about finding the angles in hyperbolic geometry and I am trying to understand this example given in Stahl's Introduction to topology and geometry. You can notice that there is a ...
3
votes
2answers
60 views

Dihedral angles of a pentakis dodecahedron

I'm new to the world of mathematical descriptions of polyhedra, and I'm wondering if, for a Pentakis Dodecahedron, the dihedral angles are uniform at each vertex. The visualization of the P.D. on the ...
1
vote
2answers
17 views

Barycentric Coordinates of Orthocenter question

this page describes the barycentric coordinates of the orthocenter as $(\tan A : \tan B : \tan C)$. How would you prove this using the areal definition of barycentric coordinates? Thank you. EDIT: ...
1
vote
3answers
57 views

Find circle radius by given triangle inside

So the triangle inside the circle: $AB = 9$cm $CB = 6$cm $CH = 5$cm I think solving this problem involves similar triangles. Thanks in advance, I'd like to have a solution suitable for 9th ...
1
vote
0answers
23 views

How to write explicity a curve on $S^n$?

I considered the $n$-sphere $S^n=\{x\in \mathbb{R}^{n+1}| \space ||x||=1 \}$ and $p\in S^n$. I want to write down explicity a curve $\sigma$ on $S^n$ passing through $p$ (for example one of the ...
2
votes
1answer
26 views

Can an isometry of the hyperbolic plane that maps a circle to a disjoint circle have a fixed point?

Can an isometry of the hyperbolic plane that maps a circle (centred on the real line) to a disjoint circle (also centred on the real line) have a fixed point? By disjoint, I mean that the two circles ...
1
vote
2answers
38 views

Affine transformation that sends a conic to itself but does not preserve the focci or the axes [closed]

So I'm trying to find an affine transformation that sends a conic to itself but does not preserve the foci or the axes. I don't know if this can be done. I'm pretty sure that if it is possible then I ...
0
votes
0answers
20 views

How prove $S_{ABC}S_{XYZ}\ge S_{MNP}^2$ for an acute-angled triangle and $M, N, P$ are points from the segments $AB, BC, CA$ respectively

Let $ABC$ is an acute-angled triangle and $M, N, P$ are points from the segments $AB, BC, CA$ respectively. Let $CM\cup NP=X, AN\cup MP=Y, BP\cup NM=Z$. How prove $S_{ABC}S_{XYZ}\ge S_{MNP}^2$? ...
0
votes
0answers
30 views

How to obtain a rectangle's side's positions if its origin isn't in its middle? [closed]

Basically I have an algorithm which generates rooms and corridors randomly and each time a room is made, a new corridor is placed on $1/4$ of the room's sides and its origin point is set to that ...
1
vote
1answer
74 views

Is $a \sin x + b \sin y \leq \sin(ax + by)$ true?

Studying math essay exam, I saw the following strange formula $$ a \sin x + b \sin y \leq \sin(ax + by), $$ where $x, y$ are arbitrary angles and $a + b = 1.$ Is the above inequality true, and can it ...
0
votes
3answers
35 views

How can I find the minimum value of this expression?

A straight line $L$ with negative slope passes through point $(8,2)$ and cuts the positive coordinate axis at $P$ and $Q$. As $L$ varies, what is the absolute minimum value of $OP+OQ$? ($O$ is ...
0
votes
2answers
33 views

Geometric and algebraic aspects of geometric vectors

I'm writing some notes for a honors physics class and I am having some trouble with some proofs. Say $\vec{A}$ and $\vec{B}$ are some geometric vectors. Then we defined the dot product ...
1
vote
2answers
55 views

Show that the Möbius band has its central circle $C$ as a deformation retract

I have started this problem by using the planar representation of the Möbius band and noted that a line down the middle is probably what is meant by the central circle, since travelling from top to ...
1
vote
2answers
27 views

Approximating length of a curved line based on Begining and End points of line

I have two points, a known distance apart. At each of these points I have a sensor that gives me flow speed and direction. I originally assumed the flow path between the first point and second point ...
0
votes
2answers
31 views

Center of mass of a wire frame?

I have this question on centers of mass which I'm trying to solve, I managed to get a value for both $x$ and $y$ of $(0.3,0.4)$ but apparently it's $0.5$ from $AD$? A uniform square frame ...
0
votes
1answer
31 views

Every tetrahedron has a vertex whose adjacent edges can form a triangle

Prove that in every tetrahedron there exists a vertex $v$ such that the three edges incident at $v$ have lengths that can form a triangle. It can be proved using a tedious casework: assume by ...
-4
votes
1answer
43 views

Find out point coordinates [closed]

In the following image all the variables are known except the point (x3,y3): How to find the coordinate of that point using other variables?
0
votes
1answer
19 views

Which points lie on the prependicular bisector of (-1,-6) and (5,-8)

$A$ and $B$ are the points $(-1,-6)$ and $(5,-8)$, respectively. Which of the following points lie on the perpendicular bisector of AB? $P(3,-4)$ $Q(4,0)$ $R(5,2)$ $S(6,5)$ Midpoint of $ AB = ...
0
votes
2answers
38 views

Find the height of statue.

Standing on one side of a 10 meter wide straight road, a man finds that the angle of elevation of a statue located on the same side of the road is X. After crossing the road by the shortest possible ...
3
votes
2answers
119 views

maximum area of semi-circle in square

I'm struggling the with the following question: Given is a square with length $a$. Now I want to find a semi-circle with the max. area. Looks like this: ...
1
vote
1answer
18 views

Extermal curve for specific problems?

I ran into a quiz question last month. how we can find the Extermal curve for following problem. $$ \int_1^2 \frac {\dot {x}^2}{t^3} dt $$ where $x(1)=2, \ x(2)=17$
0
votes
0answers
17 views

What is the isotomic conjugate version of the six point circle of isogonal conjugates?

As it is well known, the pedal triangles of a pair of isogonal conjugates in a triangle share a circumcircle. This is a nice theorem, but is there an analogous version of it for a pair of isotomic ...
0
votes
2answers
24 views

Is there a term for an “unbounded simplex”?

Is there a general term for regions like $\{(x,y):x>y\}$ and $\{(x,y,z): x>y>z\}$, i.e., regions which are simplexes with one open?
-1
votes
2answers
614 views

Find the area of this irregular octagon inscribed in a circle [closed]

Find the area of the octagon pictured here I do have some ideas how to solve it, but do not want to write them down here, because I'm hoping to find some different approaches. Also, see 1978 ...
1
vote
1answer
41 views

Is absolute value an one dimensional circle?

A circle is the set of all points that are at the same distance r from a given point in a plane (two dimensions). Similarly, a sphere is the set of all points that are at the same distance r from a ...
2
votes
1answer
30 views

Projected Area of Circle onto the Side of a Cylinder

I have encountered a problem that requires me to find the projected area of a beam of light (circular cross-section, with radius R1) vertically onto the side of a cylinder with R2 as its ...
1
vote
1answer
26 views

Dense basic open set contained in dense open subset

For an affine variety $X$ with coordinate ring $A$ it is not hard to see that for $g\in A$ the basic open set (or distinguished open set) $$D(g):=\{ P\in X | g(P)\neq 0\}$$ is dense in $X$ if and only ...
1
vote
2answers
47 views

Find distance between two poles.

2 poles, AB of length 2 metres and CD of length 20 metres are erected vertically with bases at B and D. The two poles are at a distance not less than twenty metres. It is observed that tan(angle(ACB)) ...