Questions tagged [circles]

For questions concerning circles. A circle is the locus of points in a plane that are at a fixed distance from a fixed point.

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Determine if these line integrals $\int_Cf(x,y) \;ds$ and $\int_C \mathbf{F}\cdot dx$ are positive, negative or zero. [closed]

Determine if the following line integrals $\int_Cf(x,y) \;ds$ and $\int_C \mathbf{F}\cdot dx$ are positive, negative or zero either by graphical analysis or by direct computation. $f(x,y)=\frac y {x^...
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2 votes
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Covering a square with disks

Let $d_n$ be the smallest number such that $n$ disks of radius $d_n$ cover a $1 \times 1$ square. I wonder about the asymptotics of this sequence. I was able to prove quite easily that: $$\frac{1}{\...
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0 votes
2 answers
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Find n evenly spaced points on circle with radius r

I have a circle that has a given radius $r$. I want to generate $n$ evenly-spaced points along this circle, something like this diagram (with $r=5$ and $n=4$):
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23 views

A circle and the probability as a function of time

I am trying to solve the following problem.Supposing that we have a circle of radius R which at t=0 does not contain any sphere as we had set a border so that no sphere was allowed to enter in it.At t=...
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1 vote
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Given Circle $E$ inscribed in square $ABCD,$ prove that its radius is equal to half the length of the square's side length.

Given Circle $E$ inscribed in square $ABCD$, prove that its radius is equal to half the length of the square's side length. I'm stuck on this proof but I've got a few ideas. If anyone could help that ...
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0 answers
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$z_0+r*e^{2\pi imt}$ meaning of $m$

I know that $f:[0,1] \to \mathbb{C}, f(t)=z_0+r*e^{2\pi imt}, r>0, m\in\mathbb{Z},z_0\in\mathbb{C}$ parameterizes the circle around $z_0$ with radius $r$. But how does $m$ affect the ...
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0 votes
2 answers
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How can I create an Arc path from 3 point (p1-->p2-->p3) in 3D space? [closed]

I want to create an Arc path over 3 points in 3D space. I decided to draw a circle over them and then extract sector from p1 to p3 that passes through P2. the output is: enter image description here ...
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3 votes
2 answers
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How many circles pass through 2 points but also tangent to a given circle?

Given: a circle $O$. and $2$ points $A, B$ out of that circle. How many new circles that are tangent to circle $O$ can we form which also pass through points $A,B$. My Geometry is a somewhat rusty - I ...
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2 votes
1 answer
65 views

Finding angle in circle to produce equal areas

I have a circle that is divided into 4 quadrants with a vertical and a horizontal axis. The center of the circle (where the axes cross) is point b. The top of the vertical axis is point d. On the ...
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-2 votes
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How to find distance between circles inside a square which acts as infinite dimension [closed]

so i have this square periodic boundary and i want the distance between the two circle both directly and across the boundary considering this square as surface of a sphere so basically if a circle ...
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1 vote
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Why is it true that if stereographic projection maps open balls to open balls without assuming it's already continuous?

I've been studying stereographic projection a lot recently. I've read that when proving continuity, if it can be shown that $f^{-1}$ maps open balls to open balls, then $f$ is continuous because the ...
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2 answers
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Find the area of the shaded region in the diagram, in terms of $\theta$

Problem A company is designing a new logo. The logo is created by removing two equal segments from a rectangle, as shown in the following diagram The rectangle measures $5cm$ by $4cm$. The points $A$ ...
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2 votes
1 answer
76 views

Conjecture about areas of circular segment and polygon with equal perimeter sharing a side

I was playing around with shapes and have formed a conjecture. Length of the red circular arc $=$ total length of the $n$ green line segments Conjecture: $$\sup{\left(\frac{\text{Area}_1}{\text{Area}...
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1 answer
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Using geometry to prove that $\tan(\alpha)=\frac{\sin(\alpha)}{\cos(\alpha)},\sec(\alpha)=\frac{1}{\cos(\alpha)},...$

https://www.desmos.com/calculator/6sdbz1iahd Let $\alpha$ be the angle $XOP$. We know that: $\sin(\alpha)=PA$, and $\cos(\alpha)=PB$ $\tan(\alpha)=PC$, and $\cot(\alpha)=PD$ $\sec(\alpha)=OC$, and $\...
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1 vote
2 answers
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Scale intersecting circles fixed at pivots so that they have only one point in common

Given two points, A and B; Given two circles, having 2 points in common, I1 and I2: one circle at center C1, with radius r1, with the point A on to it and another circle at center C2, with radius r2, ...
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1 answer
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Finding mean and variance of the circumference of a circle.

I was tasked with solving the following problem. Suppose that we draw a circle whose radius $R$ is uniformly distributed on the interval $(0, 1)$. Find the mean and variance of the circumference of ...
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Determine x-coordinate of 3rd point on CIRCLE from y-coordinate and 2 other points

I'm trying to construct an arc for a custom engineering curve. Say that the arc is on a circle having points $A: (x_1, y_1)$, $B: (x_2, y_2)$, and $C: (x_3, y_3)$, where $x_1 < x_2 < x_3$ and $...
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4 votes
1 answer
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Prove that there is a circle containing exactly $2018$ points

Problem Given a set $\mathtt{E}$ containing $2017^{2019}$ points on the plane. Prove that there is a circle containing exactly $2018$ points from the set $\mathtt{E}$ (these points are on the open ...
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1 answer
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Formula of an area of sector of circle without notating a radius of a circle

The following quotes are from this book(A First Course in Calculus by Serge Lang) Suppose we are given a continuous function $$r=f(\theta)\tag{1}$$ which is defined in some interval$~a\le\theta\le b~$...
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1 answer
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Why my method of calculating of the area of a circle doesn't work?

I want to manually calculate the area of a circle using integration. I assume that the circle is divided into many triangles (the number of triangles approaching infinity). The area of a single ...
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5 votes
2 answers
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APMO 2020 Geometry Problem | Proving lines to be concurrent

PROBLEM Let $\Gamma$ be the circumcircle of $∆ABC$. Let $D$ be a point on the side $BC$. The tangent to $\Gamma$ at $A$ intersects the parallel line to $BA$ through $D$ at point $E$. The segment $CE$ ...
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What is the name of the circle theorem stating that any three non-colinear points define a unique circle?

Can anyone tell if there is a name for the circle theorem which states that any three non-colinear points define a unique circle? Or how may I reference such statement. Perhaps I am too thorough about ...
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2 answers
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Dodecagon inscribed in a circle

A regular dodecagon ABCDEFGHIJKL is drawn in a circle radius 2 and centre O. Let P be the point where AD cuts the line OB. (a) Show that triangle ∆APB is similar to ∆OAB. I have researched online but ...
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2 votes
0 answers
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Intersection of inscribed equilateral triangles in a circle [closed]

I'm trying to find a proof for this proposition, but I can't find an initial hint. Note: This proposition comes from numerical evidence (Geogebra). My first try was to use formulas from integral ...
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Property about complete cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral inscribed in a circle of center $O$ and let $AB\cap DC=\{E\}$, $AD\cap BC=\{F\}$ and $AC\cap BD=\{X\}$. Let $OE\cap FX=\{G\}$ and prove that quadrilaterals $AOGB$ ...
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Determine a curve with equation $x^2+ay^2+bx+cy+d=0$ that has the highest possible order of contact with $y=cos(x)$ in $(0,1)$

Determine a curve with equation $x^2+ay^2+bx+cy+d=0$ that has the highest possible order of contact with $y=cos(x)$ in $(0,1)$ Can somebody help me, I tried to take the derivative of $y = cos(x)$ ...
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0 votes
1 answer
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Do 2 of the same circles contain each other? (Proving / Disproving)

Two Circles, $C_1$ and $C_2$ have the same center $(h, k)$ and radius $r$. Trying to prove or disprove that $C_1$ contains $C_2$ and $C_2$ contains $C_1$ OR neither contains the other.. but I think I'...
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0 answers
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Perimeter of Inscribed Square - Paradox?

Imagine a simple X / Y coordinate graph. A circle surrounds the point of origin. Let's say the radius = 3. We want to know how many points exist on the circumference of the circle through which a ...
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1 vote
1 answer
68 views

Find the value of $n$ for these circles.

Circle $B$, which has radius $2008$, is tangent to horizontal line $A$ at point $P$. Circle $C_1$ has radius $1$ and is tangent both to circle $B$ and to line $A$ at a point to the right of point $P$. ...
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0 votes
0 answers
24 views

Finding the foot of perpendicular of a point on the circle

P is a point on the circumcircle of a triangle ABC. Foot of perpendicular from P on AB and AC are (0, 0), (3, 3) respectively. If the foot of perpendicular from P to BC (whose equation is 3x + 2y + 5 =...
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1 vote
1 answer
63 views

Is my construction of one-point compactification of $\mathbb{R}$ correct? (+ Clarifying questions)

Could you please check if this construction makes sense and answer to questions to the parts in bold? My construction: The construction can be given explicitly as an inverse stereographic projection. ...
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0 votes
1 answer
58 views

Two horocycles with the same center are equidistant.

We say two circles $C_1, C_2$ are equidistant if $d(p,C_2)=K$ for any $p \in C_1$ and $d(q,C_1)=K$ for any $q \in C_2$, where $K$ is some constant. I was told that if I use the Upper-Half plane model (...
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1 vote
1 answer
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Two circles touching internally at O. A straigh line cuts these circles at A, B, C and D respectively. Prove that AB: CD = (OA × OB): (OC × OD).

S1 and S2 are two circles touching internally at 0, with S₂ being the inner circle. A straigh line cuts S1 at A, D and S₂ at B, C. Prove that AB: CD = (OA × OB): (OC × OD). My work: I found that ...
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1 vote
0 answers
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Is there a closed formula for Vonnegut's "chronosynclastic infundibulum-ated" Winston Niles Rumfoord?

In The Sirens of Titan, a major plot point surrounds the astronaut Winston Niles Rumfoord who purposefully steered his spaceship into a "chronosynclastic infundibulum" and consequently was ...
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0 votes
3 answers
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Parametrizing a circle in a counterclockwise direction

How do I parametrize a circle in a clockwise direction? For instance, if the circle is in a counterclockwise direction, the parametrization would be $$c(t) = (r \cos t,r \sin t).$$ I've seen a lot of ...
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2 votes
1 answer
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why does the area "inside" the graph of $y=\frac{\sqrt{r^{2}-x^{2}}}{\sin x}$ look like a circle

I decided to graph this ratio because both functions have to do with circles and thought the graph may look interesting, but I can't connect the dots.
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1 vote
2 answers
86 views

Given the period lengths for the orbits of $n$ different planets around the sun, how long until they all align?

Say you have $n$ planets orbiting around the sun, where the $i$th planet takes $t_i\in\mathbb{R}_{>0}$ days to complete one full cycle. Assume at $t=0$, all the planets are aligned with the sun. ...
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0 votes
1 answer
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If $z=e^{i\theta}$ is the unit circle what is $z=1-e^{-i\theta}$?

So obviously $z=e^{i\theta}$ is the unit circle. However I am trying to imagine what $z=1-e^{-i\theta}$ will look like. I know that the $1$ will shift it to be centered at $1$ not at the origin but I ...
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1 vote
2 answers
39 views

Find theta given point on circle

I am having trouble visualizing and understanding how you might obtain an angle given a point on a circle. I have a $(x, y)$ point where the values range between $0,1$ for both $x,y$. How would I ...
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0 votes
1 answer
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How to create a specific ratio in a chain of gears with fixed locations?

On a number line, there are $n$ pegs at integer positions $a_0, a_1, ..., a_n$ where $a_n > a_{n-1}$ and $a_0 = 0$. On each peg, there is a gear with a radius that can be adjusted to any positive ...
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  • 103
0 votes
1 answer
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Can a circle be drawn to pass through a point and contain a rectangle?

Let $a, b, c, d$ be points in the Euclidean plane. Suppose that $abcd$ is a non-degenerate rectangle, and that the length of the line segment $ab$ is at least as big as the length of the line segment $...
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Connected rates of change with Area and feta (Further math question)

Question about the rate of change in delta A Hi can anyone explain to me how to do section b of this question? This question is about arcs length, area of sectors and connected rates of change.
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2 votes
1 answer
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Are a circle and an ellipse isometric as smooth manifolds? [duplicate]

It is a well known result that all compact $1$-dimensional smooth manifolds are diffeomorphic to a circle. I'm wondering if the result still holds if we replace the word diffeomorphic with isometric. ...
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1 vote
1 answer
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Can a circle encompassing a continuous, closed plane curve be made to pass through an arbitrary point on the exterior of the curve?

Let $J \subseteq \mathbb{R}^2$ be a continuous, closed curve (closed in the sense that $j(a) = j(b)$ for some continuous parameterization $j:[a,b]\rightarrow\mathbb{R}^2$ of $J$ with $a,b \in \mathbb{...
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1 vote
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Area under curve in two different ways

I am currently trying to verify the computation of following integral using two different substitutions. The area below $f$ on the unit the circle with $$f(x,y) = \frac{x^2 + y^2}{4} + \frac{xy}{2} $$ ...
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  • 2,094
0 votes
2 answers
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Finding arc length without radius

Hello geniuses of the math-ternet, a 10th grade student I tutor in math showed me this exam question (ignore the scribblings). It asks for the arc length of RST, even though no radius, circumference, ...
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1 vote
0 answers
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Trying to calculate how much the Space Station can see of Earth analytically

I'm trying to calculate how much the Space Station can see of the Earth from upthere using functions and calculus. Here's the information needed: Earth's radius: $r = 6400$ ISS altitude from Earth's ...
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1 vote
2 answers
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Inscribe 3 equal circles in a Reuleaux/curved triangle without overlap [closed]

How exactly can I inscribe 3 equal circles into an equilateral Reuleaux triangle like the attached image without any overlap? I have the construction of the Reuleaux triangle down, drawn the altitudes,...
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  • 21
1 vote
1 answer
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Find the area bound by two intersecting circles and a tangent line to one of the circles

There is a circle with radius r1 and centre D This intersects a circle with radius r2 and centre C Tangent line AB is always tangential to circle with centre D It can be assumed the circle with centre ...
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0 votes
2 answers
26 views

Determining the equation, center and radius of a circumference using three points

Question We are told that a circle has the following 3 points lie on its circumference: $A=(-1,-3), B=(0,-1)$ and $ C=(-1,1).$ I have this polynomial system. In order to complete the problem, we must ...
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