Questions on the circle, a curve composed of points in a plane that are at a fixed distance from a fixed point.

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1answer
41 views

Show that a polar equation describes a circle

I want to prove that this polar equation: $$r^2 + 2r(\cos(\theta) - 3\sin(\theta)) = 4$$ describes a circle. I tried converting the equation into a cartesian equation and got $$r^2 + 2x - 6y = 4$$ ...
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1answer
25 views

Find the location of the center and the radius of the following circle: [on hold]

Find the location of the center and the radius of the following circle: $$ \left| \ \frac{z-1}{z+1} \ \right| \ = \ 3 \ \ . $$ $ \ z \ $ is a complex number. Thanks in advance!
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0answers
28 views

Finding common tangents

Known fact: For any circle with the equations $$(x-1)^2 + (y-b)^2 = r^2$$ and for any real $m$, the straight lines $$y-b = m(x-a) \pm r\sqrt{1+m^2}$$ is tangent to the circle. Question: ...
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0answers
7 views

Kiselev's Book I Plainimetry Question 242 - Question in the Description

Two lines passing through a point Μ are tangent to a circle at the points A and B. Through a point С taken on the smaller of the arcs AB, a third tangent is drawn up to its intersection points ...
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1answer
45 views

Geometry question, prove that $\angle APB = \frac12 (\angle AMB + \angle CMD)$

I got the following question: Prove that $\angle APB = \frac12 (\angle AMB + \angle CMD)$, with the following figure given: Also, the following information is given: $M$ is the centre of the ...
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1answer
36 views

Maximum number of equilateral triangles in a circle

I am stuck with a question. Given a circle with radius $x$ cm, what is the maximum number of equilateral triangles of side length 1 cm that can fit in the circle without overlapping or ...
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0answers
10 views

polar moment of area for nonplaner circle (cup)

Can somebody tell me the polar moment of area of chord for a sphere. for example when you cut a sphere at a point other than from center? Also polar moment of area for curved axis symmetry ?
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1answer
42 views

Finding all intersecting circles of one circle.

I have one circle $C_0(x_0,y_0,R_0)$ in a plane (which moves around here and there). There are many other circles on the same plane $C_1(x_1,y_1,R_1),C_2(x_2,y_2,R_2).....,C_n(x_n,y_n,R_n)$ where ...
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0answers
9 views

Shear Stress of Circular non-planer plate

Shear Stress of plan circular plate is given by T = M/($2*t*$Pi*r^2) What will be Shear stress of non-Planer circular plate? For example chord of shpere or any other circular plate but curved in ...
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2answers
46 views

Two circle intersection: help on understanding a specific explanation

As someone with basic algebra knowledge, I am having trouble understanding Paul Bourke's explanation on "Intersection of two circles" on this page. The specific part that I don't understand is where ...
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1answer
33 views

Surface area of circle extracted from a tube wall

I have made a hollow tube (thickness $1$mm) having inner radius $89$ mm and outer radius $90$ mm (length $400$ mm, can be higher). then I made a circular (circle radius $25$ mm) cut perpendicular to ...
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1answer
30 views

Finding a point along a circle a certain distance away from another point [closed]

How do I find the point(s) C (and C') which: lies on a circle centered at a point B with radius r is at distance d from point A A specific case would be: A = (0,0) B = (5,7) r=5 d=5
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0answers
30 views

How many discs necessary to cover a big circle?

Let be a circle a radius R,and other discs of radius r,palpable. I can cover the circle,completly, with a minimum number of discs,N.I can't cut any disc. What is the value of N,according to R and r? ...
5
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1answer
61 views

Arranging circles around a circle

$N$ circles are given by their radii: $r_1$, $r_2$,..., $r_N$. They are arranged around another circle so that they pack, like in this picture (order of $N$ circles should be preserved): What is ...
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1answer
39 views

How can I find the circumference of a circle using optimization? [closed]

I need Help ASAP!!!! I have a circular gutter which has a sections of the circle taken out. the total area of the circle is 8600mm^2 and i don't know how to get find the perimeter or circumference of ...
5
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4answers
152 views

Let $ABC$ be an acute angled scalene triangle.

Let $ABC$ be an acute angled scalene triangle. Let $P$ be a point on the extension of $AB$ past $B$, and $Q$ a point on the extension of $AC$ past $C$ such that $BPQC$ is a cyclic quadrilateral. Let ...
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2answers
41 views

Simple question about circles.

Let's say I have a set $S=\{(x,y): x^2+y^2=1\}$. I want to prove that for every $i \in [-1,1]$ there's a point $(i,y) \in S$. I know this sounds pretty trivial, but I need this fact for a another ...
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2answers
45 views

A Part of a semicircle between the two legs of a right angle triangle

In a right angled triangle, a semicircle is drawn such that its diameter lies on the hypotenuse and its center divides the hypotenuse into two segments of lengths 15 and 20.Find the length of the arc ...
2
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1answer
52 views

Prove that $\frac{1}{r_1}-\frac{1}{r_2}=\frac{2}{a}+\frac{2}{b}+\frac{2}{c}$

If $a,b,c$ be the radii of three circles which touch one another externally,$r_1$ and $r_2$ be the radii of two circles that can be drawn to touch these three,prove ...
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1answer
33 views

Prove that the length of the common chord is $\frac{2ab\sin \theta}{\sqrt{a^2+b^2+2ab\cos \theta}}$

Two circles ,of radii $a$ and $b$,cut each other at an angle $\theta.$Prove that the length of the common chord is $\frac{2ab\sin \theta}{\sqrt{a^2+b^2+2ab\cos \theta}}$ Let the center of two circles ...
2
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1answer
31 views

How is this circle inversion formula calculated?

I know about the inversion of a point inside a circle. But I was reading Peter Sarnak's paper on the Apollonian gasket, and got to the part where he was trying to prove descartes circle theorem. He ...
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1answer
23 views

Can someone explain this unit vector calculation for this circle inversion formula derivation?

I'm really stuck. I'm learning about circle inversion. More specifically, I was trying to understand how to derive the inversion formula for a circle, which seems to be explained here. ...
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3answers
40 views

Proving a ratio that has a relation with the Perpendicular bisectors and circumcircle

$ABC$ is a triangle, $D$ is a point on the side $BC$ of $\triangle ABC$, $R_b$ is circumradius of $\triangle ABD$ , and $R_c$ is the circumradius of $\triangle ACD$. Prove that $$ {Rb\over Rc} ...
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1answer
29 views

Prove that distance of $P$ from either of the points of contact is $\sqrt{\frac{abc}{a+b+c}}$

Three circles of radii $a,b,c$ touch one another externally and the tangents at their points of contact meet at a point $P$.Prove that distance of $P$ from either of the points of contact is ...
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1answer
67 views

Triangle construction procedure

Two lines $L1,L_2$ pass through a common point $O. $ $L_2$ goes through points $P$ and $Q$. How to construct a circle through $P,Q$ to be tangent to $L_1?$ In a particular case, at the tangent ...
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1answer
57 views

Find the area of the shaded section on a square.

In the diagram,the curved paths are arcs of circles centered at vertices $A$ and $B$ of a square of side $6$. Find the area of the shaded section $BCD$. I've been stuck on this problem for days. I ...
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2answers
44 views

Use calculus to find the area bounded by the circle $x^2+y^2-2x-2y-23=0$ and the pair of lines $x^2+2xy+y^2-7x-7y+12=0$

Use calculus to find the area bounded by the circle $x^2+y^2-2x-2y-23=0$ and the pair of lines $x^2+2xy+y^2-7x-7y+12=0$. I tried to solve the two equations by subtracting them. $2xy+35=5x+5y$,on ...
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0answers
12 views

Show that Brownian motion on the unit circle is exponentially ergodic and has the uniform measure as its invariant distribution.

My search results keep bring up planar Brownian motion on the unit disk. However, I am specifically referring to $e^{jW_{t}} = [\cos(W_t),\sin(W_t)]^{T}$ where $W_t$ is Brownian motion. I am at a ...
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2answers
33 views

How to express $\phi$ in terms of $R\text{, }x\text{ and }\theta$

Let $S$ be a circle with radius $R$ and center at $O$. Let $P$ be any arbitrary point inside circle such that its distance from $O$ is $x$ and the ray $\overrightarrow{OP}$ cuts the circle $S$ at ...
1
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1answer
24 views

How to get circle points in 3d given a radius and a vector orthogonal to the circle area?

I already know how to get a point on a circle (here), but I need a circle in 3d which should be the orthogonal to a given vector. I got: Angle in degree/radians Circle radius Orthogonal vector I ...
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5answers
54 views

Dartboard puzzle.

Given a dartboard of radius r and infinite darts.How many minimum darts you need to throw so that you can be sure that the next dart you throw is strictly less than r distance from some previous dart? ...
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2answers
66 views

Angle between segments resting against a circle

Motivation: A couple of days ago, when I was solving this question, I had to consider a configuration like this Now, I didn't intentionally make those two yellow bars stand at what appears to be a ...
2
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1answer
142 views

Sum of radii of intersecting circles internally tangent to another circle

Let $T$ be a circle with centre at $O$ and radius $R$. Two other circles $T_1$ and $T_2$ with centres at $O_1$ and $O_2$, respectively, are tangent internally to $T$. $T_1$ and $T_2$ intersect one ...
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1answer
10 views

Finding formula for set of ellipses

I'm looking for a formula for a set of ellipses lying on the intersections of two set of circles. The python code for the two sets of circles is as follows: ...
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1answer
38 views

Fraction of circle contained in another circle passing through its center.

Consider a circle $C$ with radius $r$. Now take any point on the boundary of the circle, say $P$, and draw another circle $C'$ with $P$ as the centre and radius $k\cdot r (0\le k \le 2)$. Now what is ...
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2answers
27 views

Intercept made by a line between two concentric circles

Let $$x^2+y^2-9=4r^2\enspace (r=1,2,3)$$ be $3$ concentric cirlces. Prove that the intercept made by line $$3x+4y+15=0$$ between any two cirlces is same. I thought of calculating the intercept ...
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1answer
39 views

How to calculate a, b, center of a ellipse with given bonding box of an arc [closed]

How to calculate the ellipse a, b, center, if only the bounding box of an arc, a start point and end point given. For example: (the angles are right = 0°, left=180°, top: minus, bottom:plus) Ellipse: ...
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0answers
126 views

Please help me this hard circle geometry question [duplicate]

Let $T_1$ be a circle with centre at the point $O$ and radius $R$. Two other circles $T_2$ and $T_3$ with centres at $O_2$ and $O_3$ respectively are tangent internally to $T_1$ and meet each other ...
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2answers
48 views

Geometric property Power of circle

A circle with center C is cut by a line through origin O at P and Q. If M is mid-point of PQ, show that $ OM^2 - MP^2 $ is constant for all inclinations of $OP$ and equals its power ( square of ...
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0answers
50 views

Integrating a prob distr over the set of possible circles within an annulus

Let $z$ be the measured coordinates of a point on a circle $c$ with center $x$ and radius $r$. Assume the probability of measuring $z$ given the circle $c$ is normally distributed by the distance ...
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2answers
28 views

circle tangential to inside of two intersecting circles

I need a way to find the center of a circle of a fixed size nestled tangentially on the inside intersection of two other circles of fixed size and distance, as well as its points of intersection. In ...
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1answer
23 views

Move point onto circle-outline in R3

I need to do all this in $\mathbb{R}^3$ a plane by $n \cdot p = -k$ a circle within this plane by radius = $r$ and center = $c$ a point $a$ on the inside on the circle (on the plane) a direction ...
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2answers
60 views

Find the length of triangle in three intersection circles

There are three circles ($C_1.C_2.C_3$) with radius r and they intersect each other. Suppose that $d_{ij}$ is the distance between $C_i$ and $C_j$.Is there an equation to express the length of ...
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3answers
63 views

Rotating a circle around the y-axis

What is the volume obtained when a circle of radius 2 with center (2,0) is rotated about the y-axis? I keep getting 0, but that can't be right!
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1answer
49 views

Expected area of the union of two and three circles

I have done the part of two circles. Suppose that there are two intersecting circles with radius R. And let the distance between the center of two circles is D.(0$\le$D$\le$2R) The intersection of ...
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2answers
43 views

Rotate a point on a circle with known radius and position

Having a circle $\circ A(x_a, y_a)$ of radius $R$ and a point on the circle $B(x_b, y_b)$, how can we rotate the point with a known angle $\alpha$ (radians or degrees, it doesn't really matter) on the ...
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4answers
56 views

Explanation of this integral's solution

While doodling around with circles and associated geometry, I've stumbled across this integral (I'm led to believe it is correct but I have not found or created any proof): ...
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2answers
74 views

Compute the integral over the volume of a torus,

In $\mathbb R^3$, let $C$ be the circle in the $xy$-plane with radius $2$ and the origin as the center, i.e., $$C= \Big\{ \big(x,y,z\big) \in \mathbb R^3 \mid x^2+y^2=4, \ z=0\Big\}.$$ Let $\Omega$ ...
1
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2answers
27 views

Two circles touching a line and the axes

If the circle $C_1$ touches x-axis and the line $y=x \tan\theta$,$\theta \in (0,\frac{\pi}{2})$ in the first quadrant and circle $C_2$ touches the line $y=x \tan\theta$,y-axis and circle $C_1$ in such ...
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0answers
21 views

Non intersecting chords. [duplicate]

This was a question in a math contest and it just blew me. We are given n points on circle, without any coordinates and radius. Our aim is to derive an expression in n for number of ways in which: ...