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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37 views

How to find a line cuts a circle internally?

I have a circle $S:x^2+y^2=4$ and points $A(2\cos \theta,2\sin \theta)$ and $B(2,0)$ Let $L$ be the tangent to $S$ at $A$. $C$ and $D$ are distinct point on $L$ such that $CA=AD=1$ How can I find ...
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1answer
40 views

How to find area of trapezoid formed from line connecting feet of perpendiculars to angle bisectors?

In $\triangle ABC$, perpendiculars from $A$ to the bisectors of angle $B$ and angle $C$ meet the bisectors in $D$ and $E$, respectively. The line through $D$ and $E$ intersects $AC$ at $X$ and $AB$ at ...
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197 views

Finding the Angles of a Concylic Quadrilateral

The question is as follows: Let WISH be a cyclic quadrilateral, and K be the intersection of its diagonals WS and HI. Given that arc WI is 100 degrees and arc SH is 80 degrees, find the sizes of ...
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1answer
38 views

A question on properties of triangles.

Find the relation between the circumradius of an obtuse angled triangle and its orthic triangle. I tried using angles in the orthic triangle and sine law but getting stuck. Please help me. Would ...
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2answers
37 views

Functions with other functions as their $x$-axis

I had this interesting thought today as I was playing with making images with mathematical expressions (vector images with graphing if you like) I wanted to to put a spiky thing (the sine curve) ...
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35 views

Representation of the combination of 2 circles?

I'm creating an iOS app which uses sensor and camera data, combined with GPS data, to determine a specific location for a user. I receive a new GPS location, along with a radius of accuracy, every 10 ...
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1answer
121 views

Average length of a chord in the first quadrant of a circle

One way to calculate it would be doing: $\dfrac{\int_0{}^{r}\int_0{}^{r}\sqrt{(x_1-x_2)^2+(\sqrt{r^2-x_1^2}+\sqrt{r^2-x_2^2})^2} \ \ \ dx_1 \ dx_2}{r^2}$ This seems impossible to integrate it, but ...
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1answer
76 views

Angle chasing with two tangent circles.

The bigger circle $\Omega$ is tangent to the smaller circle $\omega$. Also, $GE=2CG$. We have to find $\angle DEC$. MY WORK SO FAR. I proved using the Alternate Segment Theorem that: $$GF\parallel ED$...
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6answers
14k views

Prove the perpendicular bisector of chord passes through the centre of the circle

Hello, can someone please give me a simple proof to the following theorem: "The perpendicular bisector a chord passes through the centre of the circle." I have attached a diagram of what I mean and ...
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0answers
171 views

Circles and area of a rectangle

A circle touches two adjacent sides of a rectangle AB and AD at points P and Q respectively.Third vertex C of the rectangle lies on the circle. The length of perpendicular from vertex C to the chord ...
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1answer
188 views

Area of target board(circular shape)

I know how to find area and circumference of target boards, but I dont know why I am finding one question from new mastering mathematics for Eexcel GCSE syllabus bit weird. As the answer scheme is ...
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1answer
950 views

Using side length of triangle, find radius of touching circle

Question: The side lengths of a triangle are equal to lengths $8, 9$ and $10$. Find the exact value of the radius of the circle passing through the endpoints of the longest side and the midpoint ...
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0answers
77 views

Proof for existence of a common tangent

Can we prove that two circles touching each other always have a common tangent at the same point without using the fact that the point of contact and the centres of the circles are collinear. When I ...
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0answers
61 views

Let C and D be circles in R2 which intersect in exactly two points. Prove that iD ◦ iC has exactly two fixed points in R2 ∪ {∞}.

Let C and D be circles in R2 which intersect in exactly two points. Prove that iD ◦ iC has exactly two fixed points in R2 ∪ {∞}. I do not understand how the composition of these two become fixed ...
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1answer
489 views

Find Center of Circle given Radius, Circumference Point, and that Point's Rotation

I need to find the center point of a circle $(x,y)$ given: • The radius $\mathbf r$ of the circle • A point on the circumference of the circle $\mathbf (a,b)$ • The clockwise degrees of rotation $\...
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0answers
132 views

Coordinate geometry. Tangents and their points of contact with a circle

$ S_1 \equiv x^2 + y^2 -4x+6y -23 = 0$ $ S_2 \equiv x^2 + y^2 -4x+6y -59 = 0 $ Are two circles. $P(8,-9)$ is a point in $S_2$. The tangents are drawn from $P$ to $S_1$ and the points of contact are ...
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1answer
103 views

trisecting an arbitrary angle

I've read and seen in a video that it's suppose to be impossible to trisect an arbitrary angle with just a compass and straight-edge. This seems fairly simple though. I'm not sure I understand this ...
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0answers
106 views

Wood spring-back calculation

I am trying to make a Gothic arch greenhouse (The classic Gothic arch is known as the quinto acuto or "pointed fifth"). The arcs are 4/5 of the span. The centres of the arcs are inside the span of ...
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57 views

Orthogonal trajectories of circles with fixed points

$$C_1+\lambda \,C_2=0$$ are all circles passing through intersection of $$C_1=0,C_2=0$$ Is it possible to express orthogonal trajectories of all the $C_1+\lambda \,C_2=0 $ circles using another ...
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82 views

How is this picture regarding Möbius transformation formed?

The following picture is from the Wikipedia article on Möbius transformation. It is explained under the picture that Pre-images of the unit circle are circles of Apollonius with distance ratio $c/...
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348 views

Drawing circles enclosing a maximum number of points

Being given a radius $R$ and $N$ points on a $x-y$ plane , I want to draw a circle with radius $R$ such that it maximizes the number of points enclosed. My Approach: If there a circle ...
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101 views

What is the definition of “a circle inscribed in a square”?

I am looking for the definition of "a circle inscribed in a square" (along with any references for where the definition came from, if possible). I am also looking for any information on what is and ...
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0answers
43 views

Unit Circles in $\mathbb{R} ^2$

So I am asked to plot, for a given $p$, the $unit\space circles\space in \space \mathbb{R} ^2$. (No further information has been provided about $p$. So what I understand is that an $unit \space ...
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68 views

How do I make this equation more round?

This is my equation: $e^{-0.5*{2^y} - 0.5 *{ 2^x}}$ My goal is to create a 3D semi-circle that starts off at a corner with a z value of 1, then as it goes to the opposing corners it decays to a z of ...
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674 views

Circle with a complex radius

I am studying maths at A-level, in case that affects the level of explanations. In a Cartesian plane, a circle with centre $(a,b)$ and radius $r$ is defined as $(x-a)^2 + (y-b)^2 = r^2$. If $r^2$ is ...
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94 views

Max sum of vectors choosen from points on circles

What is an algorithm that solves this problem: Given N circles (defined by center coords and radius), what is the maximum vector sum of points, choosing one point from each circle. Note that the ...
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61 views

Is it possible to prove existence of the circle.

Is it possible to demonstrate a geometric proof to prove the existence of a circle? In Euclid's first proposition, he uses circles to prove the existence of an equilateral triangle, yet does not start ...
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654 views

Find the Radius of circle, given arc length and total change in height

Here is a picture of my problem: "m" (not sure if there is a name for this) is known and the arc length between D and B is known. BC is tangent to the circle at point B. What is the radius "r" (or ...
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1answer
379 views

How many non-overlapping circles can you fit into an area?

Say you have an area, let's use a circle, and you want to cover it with circular objects (coins for example). How many coins can fit completely into this area without overlaps or deformation the coins....
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33 views

Determine the plane which a disk occupies

I'm an EE and my last specific maths unit was a number of years ago so please forgive any incorrect terminology... I need to a way to determine the equation of the plane that a 2d circle (disc?) ...
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1answer
91 views

Given a list of the 2D coordinates of all the Pokestops in my city, how do I find a circle with radius 'r' with the most Pokestops in it?

Given a list of coordinates and a circle of radius 'r', how would I go about finding the center of the circle C in which the most points lie? My brute force solution: I have a list of coordinates (...
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144 views

How easy is this proof that the circle cannot be squared?

It is well-known that a disc can't be cut into finitely many pieces and reassembled into a square of the same area. This is Tarski's famous problem, except only allowing Jordan curve cuts. It seems ...
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1answer
152 views

Proof of Ptolemy's Theorem

I am trying to prove using similar triangles without considering an arbitrary point on the diagonals which is the most common proof what we see in text books. Let $HI=a$, $IJ=b$, $JC=c$ ,$CH=d$, $HL=...
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52 views

How to find theta between 3 circles

So here's a very ugly sketch of the question I'm trying to solve: Any assistance would be greatly appreciate! Thank you so much : 3
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596 views

Verify Green's theorem for circle centred at (x,y)=(a,0)

Verify Green's theorem on the plane for the vector field$\ \mathbf F =3xy\mathbf i -x\mathbf j$ along the circle$\ c$ of radius$\ a$ centred at $\ (x,y)=(a,0)$ with counterclockwise direction. ...
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33 views

A new line associated with a cyclic hexagon

I am looking for a proof of the problem as following: Let $A_1, A_2, A_3, A_4, A_5, A_6$ lie on a circle. Let $A_{i-1}A_i$ meets $A_{i+1}A_{i+2}$ at $B_i$. We take modulo 6. Let $O_i$ be the center ...
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1answer
270 views

12 points circle associated with a cyclic hexagon

When I research this problem A chain of six circles associated with a cyclic hexagon. I found the followings result. Let $ABCDEF$ be a cyclic hexagon. Let $A_1$ be any point on $AD$, the circle $(...
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1answer
106 views

A geometry problem hinting similarity of triangles .

I recently came across a geometry problem , published in an local magazine(publishing at high school and under graduate level) and was under Difficulty : Hard sub heading. Consider a $\triangle ABC$ ...
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1answer
85 views

Circles and generic implicit functions

I have some problems understanding circles. $x^2+y^2 = 1$ is a circle. It defines equivalence class where all (x,y) points belonging to the circle are in the same equivalence class. $(\cos a, \sin a)$...
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1answer
55 views

Geometrical question regarding two circles, a rectangle and collinear points.

Let $ABCD$ be a rectangle. Let $P$ be a point on side $BC$, and let $Q$ be a point on the diagonal $AC$. The circle through $A$, $Q$ and $D$ intersects the circle through $B$, $D$ and $P$ at points $D$...
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3answers
2k views

Finding a surface on which a given curve lies

A curve is parametrized $r(t)=(\cos t,\sin t-1,2-2\sin t), \quad 0\le t\le2\pi$ Find three different surfaces on which C lies. I have managed to find two surfaces visually: $2y+z=0$ and $x^2+(y+1)^...
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1answer
92 views

Let Q be a random point in the unit circle. Let Z=distance from Q to (-1,0) and T the distance from (1,0). What is Cov(Z,T)?

The question hints that we need to use angles since there is no joint denstity, i.e.: Let $T=h(\theta)$ and $Z=g(\theta)$ I know in general that: $$ Cov(X,Y)= E[XY]-E[X]E[Y]$$ But I'm not sure how ...
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67 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 that?...
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65 views

Prove a harmonic range from a familiar picture

During solving some simple problem (10th grade), I found this interesting problem, which I got no clue to solve it clean and properly. Hope someone can give me some hint to solve it. Thanks. Given ...
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0answers
365 views

The arc length of a circle section if radius is changing?

I would like to find the angle subtended by an arc of a circle with a changing radius. The main issue is that the radius is changing by a non-linear factor as shown below: The integral on the left is ...
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1answer
27 views

Circleline points

How can I show that $\forall$ distinct $z_1,z_2,z_3,z_4 \in \mathbb{C}$, $\frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)} \in \mathbb{R} \iff$ the points lie on a circle? I don't really have a clue of ...
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0answers
43 views

circle segment height by given fill fraction

At work I was facing the problem of how to calculate the height of a water column inside an horizontal cylinder given the volume of the liquid. A plot of this function and a visual explanation can be ...
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1answer
66 views

Formula for cycloid?

Is there a formula for cycloid? My approximation is $((2\times(x\div(\pi\div2)))-(x\div(\pi\div2))^2)^.626$.
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2answers
175 views

How to adapt “System of Circles” method to 3D for finding a sphere given 4 points?

I want to analyze (computational complexity & running time) of different approaches to determining a sphere in 3D given 4 points on its surface. To start I have been searching for different ...
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0answers
147 views

Co-ordinate Parabola Circle Contained in it; Difference in maximum and minimum possible radius

If the Difference of radii of larget and smallest Circle passing through the focus of Parabola $$Y^2=4x$$ and toughing parabola in at least one point is My Approach Let Circle be $$C: (x-a)^2+y^2=r^2$$...