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

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17
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5answers
365 views

If $A+B+C+D+E = 540^\circ$ what is $\min (\cos A+\cos B+\cos C+\cos D+\cos E)$?

Let each of $A, B, C, D, E$ be an angle that is less than $180^\circ$ and is greater than $0^\circ$. Note that each angle can be neither $0^\circ$ nor $180^\circ$. If $A+B+C+D+E = 540^\circ,$ what is ...
2
votes
3answers
139 views

Calculate the lengths of the heights descending from triangle vertices

Given a triangle, calculate the lengths of the heights descending from triangle vertices $A, B$ and $C$ respectively. $$A(-1,-1,5),~~ B(0,3,-2),~~ C(3,-1,3)$$ I don't get it with which forma i ...
0
votes
2answers
33 views

Showing that a point lies in the intersection of the closure of some subsets of $\mathbb R^d$

Let $I$ be an index set and $D_\iota\subseteq \mathbb R^d$ for $\iota\in I$ and $x\in\mathbb R^d.$ Assume that for every $\iota\in I$ there exists a sequence $(x^\iota_n)_{n\in\mathbb N}\subseteq ...
5
votes
2answers
246 views

Do I have enough iMac boxes to make a full circle?

My work has a bunch of iMac boxes and because of their slightly wedged shape we are curious how many it would take to make a complete circle. We already did some calculations and also laid enough out ...
1
vote
1answer
126 views

Convex set, differentiable in one point, inner direction

I have a convex set $\mathcal{C}$ with non-empty interior and I have shown that its boundary is differentiable in a point $c^*$, in the sense that there is a vector $n^*$(the normal vector), such that ...
3
votes
2answers
2k views

Proper Notation for Lines in Geometry

If a line is simply named with a single letter, say for example $k$, is it incorrect to refer to this line as $\overleftrightarrow k$?
2
votes
1answer
253 views

Find a projection of a $k$-simplex with minimal “radius”

Let $S$ be a $k$-simplex in $\mathbb{R}^k$. I'd like to find a hyperplane $P$ that passes through the origin such that projections of vertices of $S$ onto $P$ are as close as possible to the origin ...
0
votes
2answers
82 views

Diameter calculation

I've been killing myself with this. I got it down to one angle, but I can't determine that angle, so please, if anybody has any idea. Thanks Cylinder on the V frame. V frame has an angle of 11$^o$ ...
0
votes
1answer
216 views

Subdivided icosahedron points do not lie on circumscribed sphere

How do we subdivide an icosahedron so that the new vertices created on each subdivision also lie on the circumscribed sphere. i.e. have the same radius as the original icosahedron? If we just create ...
1
vote
2answers
370 views

find symmetric equation along line $y=-x$

generally i know that to find symmetric equation of function along line $y=x$,we should exchange $x$ and $y$ and solve back,but what about $y=-x$?should i repeat again the same procedure,but ...
2
votes
2answers
92 views

Geometry: “Repulsion” between two lines in three dimensional space

Given two points in three (or, any) dimensional space, the distance between these two can easily be plugged into a function to return a value for repulsion or attraction, ideally this function is ...
0
votes
2answers
41 views

find scalar product of vectors in rectangular

let us consider following problem and picture we have $ABCD$ rectagular with $AB=3$ and $BC=5$,$F$ and $E$ are midpoints of rectangular sides,we should find scalar product of my question is ...
2
votes
1answer
523 views

Intersection of two arcs

I have two circular arcs with 3 points. Is there any algorithm to check if the arcs intersect or not? I already found many algorithm for circles but I'm looking for arcs.
0
votes
3answers
474 views

How to check the line segment is normal to ellipse?

My line segment has one end touching the ellipse The other end of the line segment can be outside or inside the ellipse not on the ellipse The ellipse centre is in the origin Line Eqn, $y = mx + c$ ...
0
votes
1answer
142 views

Quadrangle with maximum area in circle

My question is related to my previous one find parameter for maximize area suppose we have $4$ points,with coordinate $A(2\cos t,2\sin t)$ $B(-\cos(2-t), -\sin(2-t))$ $C(-2\cos(t) ,-2\sin(t) )$ ...
5
votes
2answers
81 views

Examples where $1 \in W_0^{k,p}\left( U \right)$

$M$ is a Riemannian manifold, $U$ is a domain in $M$. Consider the Sobolev space $W_0^{k,p}\left( U \right)$: the closure of $C_0^\infty \left( U \right)$ (smooth functions with compact support) in ...
0
votes
2answers
232 views

determine position of circle inside square

i need to determine position of circle inside square,let us suppose that we have following picture we have following informations: 1.$ABCD$ is square 2.all small figures ,$KMCE$,$PKEF$,$NPFD$ ...
3
votes
1answer
980 views

parametric equations of folium of Descartes

We know the function of the folium of Descartes is $x^3+y^3=3axy$. The problem is to show that the folium of Descartes has parametric equations $x=\frac{3at}{1+t^3}$, $y=\frac{3at^2}{1+t^3}$ (this ...
2
votes
0answers
144 views

Computing distances between hyperspheres and sides of a hypercube?

Suppose you are given the $n$ dimensional hypersphere: $$\left(x_1 - \frac{1}{2}\right)^2 + \left(x_2 - \frac{1}{2}\right)^2 +\ldots+ \left(x_n - \frac{1}{2}\right)^2 = \frac{n}{4}$$ And the ...
1
vote
2answers
182 views

Fitting circle into an angle

I've been struggling with this for quite some time now, anyone could help me perhaps with this? Given an angle of an arbitrary degrees, and a circle with radius r. And imagine I would try to push the ...
2
votes
3answers
4k views

project a point in 3D on a given plane

A point in a 3D space is given as $ P(x,y,z) $. I want to find the position of this point projected parallel to the normal on a plane Q defined by $3$ non-collinear points $ Q1(x1,y1,z1), ...
1
vote
3answers
191 views

Nature of a triangle with vertices $z_1, z_2$ and $-1$ such that $|z_1|=|z_2|=1=z_1+z_2$ [closed]

If $z_1$ and $z_2$ are distinct complex number such that $|z_1|=|z_2|=1$ and $z_1+z_2=1$, then the triangle in the complex plane with $z_1,z_2$ and $-1$ as vertices must be: equilateral. right ...
3
votes
4answers
341 views

Finding the area, general case with angle $\theta$.

Inspired by this question, I am curious to know the more general case. Given the radius of the large circle as $R$ and the angle $\theta \le \pi$, what is the area of the colored section? My ...
2
votes
1answer
223 views

annihilator of an intersection in infinite dimension

Given two subspaces of an infinite dimensional Banach space, is the sum of their annihilators dense in the annihilator of their intersection?
4
votes
5answers
180 views

Minkowski sum of two disks

An open disk with radius $r$ centered at $\mathbf{p}$ is $D(\mathbf{p}, r)=\{\mathbf{q} \mid d(\mathbf p, \mathbf q) < r\}$, and the Minkowski sum of two sets $A$ and $B$ is $A \oplus B=\{\mathbf p ...
3
votes
2answers
584 views

Proof: Invariant angle measure - same result for any circle drawn.

Below I have quoted Wikipedia. I am particular interested in the statement: The value of $\theta$ thus defined is independent of the size of the circle: if the length of the radius is changed ...
2
votes
1answer
435 views

2D calculate position of a point relative to 4 known points

I have 4 known points (a square) in 2D space: A: {x:0, y:0} B: {x:100, y:0} C: {x:100, y:100} D: {x:0, y:100} Then I have a point inside the square. I don't know its location, but I do know the ...
7
votes
3answers
430 views

find area of dark part

let us consider following picture we have following informations.we have circular sector,central angle is $90$,and in this sector there is inscribed small circle ,which touches arcs of sectors ...
2
votes
2answers
2k views

Ellipse circumference calculation method?

Actually I know how to calculate the circumference of an ellipse using two methods and each one of them giving me different result. The first method is using the formula: ...
3
votes
2answers
98 views

What's the symbol m in this sum?

I'm supposed to write some code to calculate the inertia moments of a shape, but I am afraid I have been given too little information. The matrix that I must obtain is this one: $$ \begin{vmatrix} ...
8
votes
5answers
3k views

Proving the length of a circle's arc is proportional to the size of the angle

How can I prove that: The length of the arc is proportional to the size of the angle. Every book use this fact in explaining radians and the fundamental arc length equation $s = r\theta$. ...
0
votes
1answer
267 views

Prove that in an obtuse triangle the orthocentre is the excenter of the orthic triangle

Consider an obtuse angled $\Delta ABC$ with altitudes $AD, BE, CF$ concurrent at $H$. Consider the orthic triangle $\Delta FED$. Extend $ED$ to $D'$ and $EF$ to $F'$. Prove that $\angle FDH = \angle ...
3
votes
1answer
401 views

Shortest Path and Minimum Curvature Path - implementation

Let's say we are given a race track, which may be described as a closed curve of given width (it may differ along the curve). My task is to implement an algorithm which finds two kinds of trajectories ...
1
vote
4answers
5k views

Why is volume of cylinder > volume of cube

If we have a cylinder with radius 1 and height 1 and cube with side lengths equal to 1 Volume of cube = $r^3 = 1^3 = 1$ Volume of cylinder = $\pi r^2h = \pi 1^2\times1 = \pi$ clearly $\pi > 1$, ...
0
votes
1answer
84 views

Find angle in another view of image

Is it possible to find angle in some "virtual" view from scaled view? (yes it is :] ) What i mean, for example: You got table and you look at it from some position (you know that the left top corner ...
1
vote
1answer
142 views

find area of square part

let us consider following picture where $ABCD$ is square, and using $A$ and $C$ as center,there is drawn arc,we should find area of dark part.we know that length of square is $a$,as i see the ...
1
vote
3answers
2k views

Find the equation of a line given the bounded area [closed]

Find the equation of the line through $(2, 2)$ and forming with the axes a triangle of area $9$.
0
votes
1answer
269 views

find minimum length of triangle

suppose that we have $ABC$ triangle,with $AB=28$ and $C=120$,we should find minimum length of triangle,if it is know that $AC:BC=3:5$,it is clear that minimum side is $AC$,also because sides are ...
1
vote
2answers
132 views

find angle in triangle

Let us consider problem number 21 in the following link http://www.naec.ge/images/doc/EXAMS/math_2013_ver_1_web.pdf It is from georgian national exam, it is written (ამოცანა 21), where word ...
0
votes
1answer
2k views

How to find rotation angles along X,Y,Z axes with a known vector to bring the axes to correct situation

I am working with 3d point data. When I checked the data I realized that there is some error on my data and need to do some kind of rotational rectification because the points which should be ...
1
vote
1answer
64 views

Trapezoid, find the sides

I have a right tangential trapezoid. I know the radius of the circle inscripted and the perimeter of the trapezoid. How can I find the sides?
0
votes
1answer
418 views

What is the height of a regular polygon?

I have three small circles forming a pyramid. I would like to centre that group in a square but have spent a couple of hours trying to calculate the height of the pyramid. I just can't seem to get ...
2
votes
3answers
1k views

Visual proof of the addition formula for $\sin^2(a+b)$?

Is there a visual proof of the addition formula for $\sin^2(a+b)$ ? The visual proof of the addition formula for $\sin(a+b)$ is here : Also it is easy to generalize (in any way: algebra , picture ...
-1
votes
1answer
79 views

Internally diving of vectors

Given the vectors $$\begin{eqnarray*}A&=&i+j-k\\B&=&i-j+2k\\C&=&j+k\end{eqnarray*}$$ How do I find the position vectors which divide BC AC internally in the ratio of 3:2?
2
votes
3answers
1k views

prove that minor arc of a great circle is the shortest distance

How to prove that the minor arc of a great circle is the shortest path connecting two points on the surface of a sphere? The key to solving the problems is that we must take all curves connecting ...
1
vote
5answers
2k views

Finding perpendicular bisector of the line segement joining $ (-1,4)\;\text{and}\;(3,-2)$

Find the perpendicular bisector of the line joining the points $(-1,4)\;\text{and}\;(3,-2).\;$ I know this is a very easy question, and the answer is an equation. So any hints would be very nice. ...
2
votes
2answers
268 views

Calculate the inner angles of the triangle $A(2,-3,5),B(0,1,4),C(-2,5,2)$

I want to calculate the inner angles of this triangle. $$A(2,-3,5),B(0,1,4),C(-2,5,2)$$ I know that for calculate the angle I need to do the following thing: $$\cos(\alpha)=\frac{A\cdot B}{|A||B|}$$ I ...
1
vote
1answer
88 views

Will a point moving on a sphere always at an angle x (0 deg. < x < 90 deg.) to the “equator” reach a “pole”?

Formulating my question seems to have given me the answer: that the point will continue getting closer to the pole but never reach it. Am I correct? Edit in response to Martin Argerami: I see ...
4
votes
2answers
187 views

A proof in circles.

I need help proving this problem: $AB$ is a diameter of a circle. $CD$ is a chord parallel to $AB$ and $2CD = AB$. The tangent at B meets the line $AC$ produced at $E$. Prove that $ AE = 2AB $. ...
0
votes
1answer
58 views

Generalizing a statement about points in the unit square

What is the three-dimensional version of this statement: Any $n$ points in the unit square can be labeled $x_1,\ldots,x_n$ to satisfy the inequality $$\|x_1-x_2\|^2 ...