For questions conserning circles. A circle is a curve composed of points in a plane that are at a fixed distance from a fixed point.

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3
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1answer
68 views

$\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ - Theory of distribution

An exercise asks to find the wave trace $w(t)=\operatorname{tr} \left(e^{it \sqrt\Delta}\right)=\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ as a distribution (or generalized function) of the Laplacian ...
1
vote
0answers
77 views

$\langle w,\varphi\rangle =\int_{\mathbb{S}^1} \left(\sum_{k \geq 1} e^{itk}\right) \varphi(t) \, dt$ - Generalized function

An exercise asks to find the wave trace $w(t)=\operatorname{tr} \left(e^{it \sqrt\Delta}\right)=\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ as a distribution (or generalized function) of the Laplacian ...
0
votes
5answers
41 views

Finding a Coordinate on a circle using radius, angle, and origin

I am trying to calculate a point on a circle using an angle and a different point. With this picture, I know the origin O, the radius r, the angle A, and the point B. Now I want to find the point ...
0
votes
2answers
54 views

A triangle in a circle

According to the following picture $E$ is the midpoint of $BD$ and $DC=BD$. If measure of $\angle EGF$ is equals to $90$ degrees then find the value of $\frac {DE} {EF}$.(point A is the center and BC ...
0
votes
1answer
41 views

Circle tangent to three tangent circles (without the Soddy/Descartes formula)

We have three circles tangent to each other with radii $1$, $2$, and $3$. Another circle is tangent to the other circles; find the radius of that circle using elementary geometry, without the Soddy ...
0
votes
2answers
28 views

Need help with alternative method to equation of a tangent at the point of a circle

so I know a simpler of looking for the equation of a tangent at the point of a circle is to differentiate, my lecturer would rather we not use calculus and has charged us with looking for an alternate ...
1
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2answers
50 views

Why does convention not recognize $2\pi$ as the fundamental quantity? [duplicate]

It looks as though my question might turn out to be a duplicate. If so, it does not need an answer, after all, thanks. ORIGINAL QUESTION Why does convention recognize $\pi\approx 3.14$, rather than ...
1
vote
3answers
23 views

Process for solving this system of equations

I have this system of equations for which I'd like to solve for $x$,$y$, and $r$ where $a$,$b$, and $t$ are constants: 1: $0 = (x-a)^2 + (y - b)^2 - t^2$ 2: $y = \dfrac{bx-rx+ar}{a}$ 3: $r = ...
0
votes
1answer
20 views

The Locus Of M (Repeated Questuon) [duplicate]

Let A and B be two fixed points on a straight line. Two circles touch this line at A and B respectively and the tangent to each other at M, when the circles vary the locus of M is? This question has ...
0
votes
0answers
18 views

Increasing or decreasing theta based on direction of vector

I am a programmer, and I have some holes in my math knowledge that I am working on filling in. Right now I'm working with a simple process involving drawing curves and straight lines. The line that is ...
1
vote
2answers
35 views

Show that there are at most two rational points on $(x - a)^2 + (y - b)^2 = r^2$ for $a, b$ irrational.

For any given irrational numbers $a, b$ and real number $r \gt 0$, show that there are at most two rational points (points whose coordinates are both rational numbers) on the circle $(x - a)^2 + (y - ...
1
vote
2answers
25 views

Converting the Great Circle distance to direct distance between two points on earth?

Apologies if this question has been asked before. Across the surface of the Earth, the distance between London and New York is 5567 km. Given that the earth has a radius of 6371 km, what is the ...
0
votes
3answers
45 views

Prove that : A circle consist of infinite points

How to prove a circle consist of infinite points ?Proof using calculas or computational theory is appreciated?
0
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2answers
50 views

Find nearest points on the circumference of a circle based on reference coordinates and centre coordinates given

For a programming purpose I've been asked to plot few points next to a point on a circular diagram. The only given values are the reference point coordinates and distance from/to of the new point to ...
2
votes
0answers
37 views

Volume of “the complex projective space” of a certain radius.

Consider the circle action on $\mathbb C^n$ given by $(e^{it},z)\to e^{it}z$. A moment map for this action is $J:\mathbb C^n\to\mathbb R:z\to -\frac{1}{2}|z|^2$. Let $M_l=J^{-1}(-\frac{l}{2})/U(1)$ ...
0
votes
2answers
122 views

Wave kernel for the circle $\mathbb{S}^1$ - Poisson Summation Formula

Question : How could I compute the (wave) kernel from the fact I have already found (wave) trace on unit circle? The definitions are related to the page $25$ of the following pdf. As the ...
2
votes
2answers
106 views

Minimum Area of An Ellipse Surrounding Four Circles

The circles are all four combinations of $(x\pm60)^2+(y\pm25)^2=5^2$ (see pic at end). The ellipse I've got is one I found via trial and error but there must be an analytical way to solve this, ...
0
votes
1answer
35 views

Cirle's Center and Radius for Lots of Point

I know that If I have 3 points I will have this center (I calculated this) a=\left[\begin{matrix}x1^2+y1^2&y1&1\\x2^2+y2^2&y2&1\\x3^2+y3^2&y3&1\\\end{matrix}/ ...
0
votes
1answer
31 views

Geometry including semicircle and arc length

This is a question in the Princeton online test of GRE general book. I got it wrong, but even when I looked at the answer I find it difficult to understand how the answer is obtained. In the ...
0
votes
1answer
54 views

Sine law and circumscribed circle

How is $\frac{a}{\sin(A)}=2R$ (where $R$ is the radius) derived?
0
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0answers
32 views

Finding the Centre of a Circle using Software [closed]

I have the following image, and want to pinpoint the exact pixel-coordinates of the centre of the large circle in the middle. Is this possible using any freely available software?
0
votes
2answers
40 views

Putting Numbers on a Circle

If I have a circle and I start numbering points along the circumference with all the natural numbers: 1, 2, 3, 4, and so on, such that the length of the arc between two consecutive numbers is ...
4
votes
3answers
70 views

Right Triangle and Circle Theorem

Let $ABC$ be a triagnle such that $\angle BAC$ is a right angle. Suppose $D$ is a point lying on $BC$ such that $BD=1$, $DC =3$ and $\angle ADB=60^{\circ}$, find the length of $AC$. I was told that ...
1
vote
3answers
61 views

Angle between squares at which they just touch along the circumference of a circle

Say I have two squares whose centers fall along the circumference of a circle. The circle has radius $x$. The squares have the same height and width $y$. The height of one square is parallel to the ...
3
votes
0answers
31 views

Why does (h,k) generally represent the center of a circle?

Why are h and k generally used to denote the coordinates of the center of a circle? After a bit of research, we found that h may represent "horizontal shift" or "horizontal translation", but we're ...
1
vote
0answers
50 views

Spectrum of the circle - Clarification of certain points

I have a bit of difficulty as doing the problem Eigenvalues of the circle over the Laplacian operator. Since $g$ is a periodic function, do I have to use the Fourier series? If so, how could I do ...
7
votes
3answers
63 views

Number of rectangles to cover a circle

After searching around I found this is similiar to the Gauss Circle but different enough (for me anyway) that it doesn't translate well. I have a circle, radius of 9 that I need to completely cover ...
-1
votes
2answers
32 views

Finding a length (x) inside a circle sector given another length (y) and the arc length (s) [closed]

I am stuck on a problem and can not seem to find a solution, maybe someone here can help me or at least tell me if it is possible to solve? Please look at the figure: The problem is: Find the ...
2
votes
1answer
29 views

Questioning about the meaning of “$1$-dimensional circle”

When we talk about the $1$-dimensional circle, is it a one-dimensional object, although one can embed it into a two-dimensional object? More precisely, is it a one-dimensional manifold?
9
votes
2answers
88 views

How many mutually orthogonal circles are possible?

How many mutually orthogonal circles is it possible to have? It is easy to construct $3$ mutually orthogonal circles, e.g. $3$ circles with radius $1$ and centers at the vertices of an equilateral ...
2
votes
1answer
22 views

Find radius of a circle from intersecting chords

Say I have two chords that intersect inside a circle, not at a right angle, and neither is the diameter. It seems to me this is enough information that the circle must be unique, but I can't seem to ...
0
votes
1answer
21 views

Problems on measure of angles and arcs in a circle diagram

A friend of mine recommended this site. I cannot figure out any of the parts in the problem in the picture click here The line segments AE and DE are not tangent to the circle, so I don't know where ...
1
vote
3answers
42 views

Same perimeter and area for a circle and an ellipse

For a given circle, is there exist an ellipse with same perimeter and area as to that circle? If not, that is my suspicion, is in three-dimension parallel question: For a given sphere, is there ...
-1
votes
0answers
32 views

Circles in k-connected graph

Can somebody help me with this question: Does there exist a $k \geq 4$ such that for every $k$-connected graph $G$ there is a set of its vertices $v_1,v_2...v_k$ for which there exists a circle that ...
1
vote
1answer
24 views

Find altitude of equilateral triangle given inscribed circle dimensions and position

I've found myself trying to solve this for my Geometry class where we have to model a basic piece of architecture and find its volume and surface area (very basic). But the structure I chose requires ...
0
votes
0answers
12 views

compute integrals on the circle group

Let $\theta(t)\in SO(2)$, where $SO(2)$ is the special orthogonal group. I want to compute $\theta(t)$ by integrating an 'angular velocity', say $\omega(t)\in\mathbb{R}$. Hence, I want to write $$ ...
0
votes
1answer
21 views

A circle tangent to two circles touching internally and line

Find the radius of a circle touching two circle $x^2+y^2+3\sqrt{2}(x+y)=0$ and $x^2+y^2+5\sqrt{2}(x+y)=0$ and also touching the common diameter of the two given circles. The two circles touch ...
0
votes
2answers
33 views

Tangent line to quarter circle inscribed in square ABCD intersects point X. Where to start?

Disclaimer: The answer must be an integer, as all competition problems were designed to yield integral answers. Hello! Yesterday I underwent a math tournament. There was one problem that was rather ...
0
votes
1answer
44 views

Circles overlapping a central point

If I have a circle x with radius r. How many circles can I add around it with same radius such that these circles overlap the center point of circle x without overlapping any other circles' center ...
2
votes
2answers
32 views

Can a polygon with an infinite number of sides be viewed as a line?

The inner angles of a polygon approach 180º as the number of sides (N) of the polygon increases. So, if N approaches infinity, we would have a circle. But... At infinity, we would also have a set of ...
2
votes
1answer
34 views

Tournament of Towns Geometry Problem, Proof by Construction?

I was to prove the following proposition from an old Tournament of Towns problems archive: Problem. A circle $\omega_{1}$ with center $O_{1}$ passes through the center $O_{2}$ of another circle ...
0
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3answers
46 views

A simple proof that a polygon circumscribing a circle overestimates its perimeter

Looking at the picture below, it's easy to see why the perimeter of a polygon inscribed in a circle is an underestimation of the circle's perimeter. This follows from the triangle inequality: Any side ...
0
votes
1answer
31 views

Given a chord length and distance from center find length of a different chord

A chord that is of length 18 cm is 12 cm away from the center of a circle. How far is a chord of length 10 cm from the center? I know that chord of equal distance away are equidistant from the center ...
0
votes
2answers
28 views

How do i compute the closest points on a sphere given a point outside the sphere?

I looking for method which can compute the yellow area in this image.. The ball with the green fill is a sphere, where i know the center point and the radius of it. The circle with the red fill ...
5
votes
3answers
49 views

Finding the Center of a circle given two points and a radius (algebraically)

Preface: I'm writing a program in which I need to find the center of a circle, given two points on the circle, and the radius. Therefore, a construction or doing the problem out by hand is not an ...
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votes
3answers
90 views

Prove that $\mathbb{S}^1$ is not homeomorphic to $\mathbb{S}^2$.

I've been able to show that $\mathbb{R}$ is connected and $\mathbb{R}^2 \backslash \{ x\}$ is connected for any $x \in \mathbb{R}^2$. Using this I was able to show that $\mathbb{R}$ and $\mathbb{R}^2$ ...
-1
votes
1answer
35 views

how to distance circles drawn on another circles

I need to do some calculations in order to do this drawing (sorry for the quick sketch): I need to define a set of variables and do simple calculations as much as possible in order to come up with ...
-3
votes
1answer
47 views

How to calculate angle within a circle

Given this situation where the circle is cut by the rectangle. How do you calculate the angle α
0
votes
0answers
42 views

Double Integral over the region of an ellipse cut off by a circle

I've been stuck on this question for awhile. I need to calculate the double integral $\iint_R \frac{1}{r^3} dA$ using polar coordinates. R is the region displayed below: The ellipse has centre ...
1
vote
2answers
63 views

Evaluate the double integral $\iint_D\sqrt{4-x^2-y^2}$ bounded by semi-circle

I would appreciate it if someone can help me solve this question, as I'm struggling to get its answer. Q: Evaluate the double integral $$\iint_D\sqrt{4-x^2-y^2}dxdy$$ bounded by semi-circle ...