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

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How to calculate angles and areas (circles)- AS Maths

Hi here's the question: A(-1,-4) and B(6,-5) are points on the circumference of a circle, centre D(3,-1). The tangents at A and B intersect at C. How would I find the angle ACB and the area of ACBD? ...
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21 views

How to evaluate solid angle subtended by a segmented circle?

The diagram above shows a circular plane, centered at the origin 'O', has a radius $7 cm$. Two identical rectangular strips, each having width $2 cm$, are thoroughly cut off from the circular plane ...
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2answers
43 views

Circle inside circle [on hold]

$24$ small circles of radius $2$ units are placed in a large circle of radius $R$. All the small circles are touching each other and the outer circle. Find value of $R$.
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17 views

Check if points are sorted in circular order

How do you check if points are sorted in circular order (regardless of clockwise or counter-) (assuming they don't exactly form one whole circle, what matters is the points are sorted in a circular ...
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0answers
32 views

Truth value of a mathematical statement about circles?

Let $A$ be the set of circles in the plane with center $(0,0)$ and let $B$ be the set of circles in the plane with center $(-2,3)$. Furthermore, let $P(C_1,C_2)\colon C_1$ and $C_2$ have exactly one ...
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1answer
39 views

Use calculus to derive area of circle using n triangles

This is a homework question I am struggling with... Let $n$ be a positive integer, and cut the circle into $n$ equal sectors. In each sector there is an isosceles triangle formed where the edges of ...
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1answer
47 views

Construction of a common mean proportional

"Given four points, A, B, C, D in order on a straight line construct a point P on BC such that PA.PB = PC.PD" I assume the end result is to have two right angled triangles AXP with X perpendicular to ...
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2answers
25 views

Caclulate X,Y coordinates of point after rotation around another point of given degrees

There are Two Points A and B. The linear distance between the points is R. I have the ...
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2answers
48 views

Four complex numbers $z_1,z_2,z_3,z_4$ lie on a generalized circle if and only if they have a real cross ratio $[z_1,z_2,z_3,z_4]\in\mathbb{R}$

Let $[z_1,z_2,z_3,z_4]$ denote the cross ratio of the complex numbers $z_1,z_2,z_3,z_4\in \mathbb{C}$. Show that the distinct points $z_1,z_2,z_3,z_4\in\widehat{\mathbb{C}}$ lie on a generalized ...
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0answers
27 views

Determine the equation of this circle. [closed]

Two lines $l_1$ and $l_2$ are given as: $$\begin{cases}l_1:~3y=-x+6\\ l_2:~2y=x+12\end{cases}\quad\textrm{and}\quad l_1\perp l_2$$ A circle of radius $5$ cm has as its centre, the point of ...
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1answer
20 views

Finding Chord Length with only points on circumference,radius and center

I have a table of sin and cos values. I know that the radius of the cirlce is 1. The center is 0. I'm trying to figure out the CHORD length between the points on the circumference. E.g. of points ...
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1answer
46 views

Calculate coordinates of a point on a circular arc

I have following dilemma I need to solve for my software: I have a circular arc, and I do know these: x,y coordinates of startpoint of the arc x,y coordinates of endpoint of the arc x,y coordinates ...
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2answers
45 views

Finding the equation of a circle given two points on the circle

11. Find the equation of the circle which touches $x^{2} + y^{2} - 6x + 2y + 5 = 0$ at $(4, -3)$ and passes through $(0, 7)$. My textbook has a worked example for obtaining the equation of a ...
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0answers
13 views

Geometery finding measures of inscribed angles [closed]

I recently flunked a geo quiz and i need to make corrections and i cant figure out even after reviewing my notes for hours. Please help
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1answer
32 views

Proof of Compound Angle from Ptolemy's Theorem

I have a query regarding a proof I'm reading on the additive Sine compound angle formula, which uses Ptolemy's theorem. http://www.cut-the-knot.org/proofs/sine_cosine.shtml I'm looking at the ...
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5answers
240 views

Technique for proving four points to be concyclic

While making my way through an exercise, I stalled on question 7: 7. Prove that the points $(9, 6)$, $(4, -4)$, $(1, -2)$, $(0, 0)$ are concyclic. The book does not provide any guidance on how ...
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1answer
41 views

Draw the line segment joining the centers of two circles. Where does it meet the circles?

I'm trying to construct a line segment between two circles. Given each radius and $x$, $y$ center of each circle, how can I find the endpoints for the blue line segment?
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0answers
21 views

Work back from atan2 please?

I have input from 2 axis which range from -1.0 to 1.0 and I convert that into degrees using the below formula. degrees = Math.toDegrees(atan2(axis1, axis2)) so an input of 1.0 and 1.0 gives a ...
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0answers
32 views

What radius circle to remove from unit circle to make golden earring?

A circular lamina of radius $x$ is removed from a circular lamina of radius $1$. If the center of gravity is at the edge of the smaller circle (along the line connecting the two centers) what is $x$? ...
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0answers
34 views

Coordinate Geometry Help (circles + trigonometry) [on hold]

Question : Find all points $(x, y)$ {if there are too many then number of points is enough} which lie on or inside the circle $x^2 + y ^2 = 9$ and satisfying the equation $\tan^4 (x) + \cot^4 (x) + 2 ...
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1answer
37 views

“Reverse engineering” of a geometric illustration

The following enigmatic illustration can be found here, unfortunately without any accompanied comment or short description: Can you deduce its meaning? What was the way it was constructed?
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1answer
28 views

Question in finding final position in circle

im having difficulty solving this problem right now, I tried couple of times but i couldnt figure it out, can you please help and explain? thank you
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1answer
28 views

Integration: Find length of curve using NINT

Here are the questions - For question 4, part (b) gives a unit circle. But I'm unable to proceed with parts (a) and (c), since the curve is double valued for -0.5 Also, for question 6, integration ...
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2answers
62 views

Covering 1/8 of a circle

A circle of radius 1 is given, and 8 semicircles of radius 1/2, like in this picture: What is the radius of the smallest circle that can cover shaded area? There was another problem involving the ...
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3answers
48 views

Inverse of an ordered pair?

Let $f: A \to B$ be a bijective function where $A = [0, 2\pi)$ and $B$ is the unit circle. Find the inverse of $f(\theta) = (\cos\theta, \sin\theta)$. I don't understand what it means to take the ...
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5answers
67 views

Find third point to make isosceles triangle with a specific area

Using points A(1,2) and B(-2,-2), find a third point, with a positive y-value, that makes ABC an isosceles triangle with area 10 units${^2}$. I have found AB to be 5 and used this as $r^2$ below.. ...
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1answer
58 views

What is the equation to evenly distribute circles in a spiral?

What is the equation to evenly distribute circles in a spiral? I have attached a picture to show what I am trying to achieve and need to know what the equation is for this.
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0answers
21 views

How are 2D collision forces calculated?

Between 2 circles of the same radii, how can I calculate the collision forces to apply to each of the 2 circles? I have position, mass, and velocity for each of the circles. Here's what I have ...
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1answer
38 views

Calculating the length of a tangent drawn to a circle from a named point

My book (New Tertiary Mathematics Volume 1 Part 1, by C Plumpton and P S W Macilwaine) describes a method for calculating the length of a tangent to a circle from the point $(x_{1}, y_{1})$ outside ...
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0answers
34 views

Formula for area of circle made up of squares

I need to draw an approximate circle on a grid of squares and find its area. Each square must either be completely part of the circle or not at all occupied. Obviously, this means that it cannot be a ...
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0answers
42 views

2015 AMC12A question 25

This is a question from the 2015 AMC12 math competition. I haven't really made much progress at all on it, and I just want to know the right way to solve this equation. A collection of circles in ...
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1answer
44 views

Does this equation for a tangent to a circle have a name?

My Maths tutor showed me a shortcut way to find the equation of a tangent to a circle, given the radius, centre and point the tangent touches the circle: $$(x - a)(c - a) + (y - b)(d - b) = r^2$$ ...
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3answers
57 views

Find circle radius by given triangle inside

So the triangle inside the circle: $AB = 9$cm $CB = 6$cm $CH = 5$cm I think solving this problem involves similar triangles. Thanks in advance, I'd like to have a solution suitable for 9th ...
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0answers
12 views

Find ArcLength given a set of circle parameters.

I need an equation to help define a 3D curve, given starting vector, ending vector, starting point, and vertical distance between start and end points. The curve should be a segment of a circle ...
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1answer
23 views

Finding coordinates of the third point of a triangle from given?

In ABC triangle we know the coordinates of A and B vertices. We also know lengths of 2 edges shown in the picture and the third edge is calculatable. What is the most efficient functon to find x3 and ...
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2answers
32 views

Circular measure

Hi everyone, This is a question from a June 1984 cambridge past paper. I'm getting stuck with the part (c) and the 'hence show...' Please someone can help, I'd be very grateful.
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1answer
49 views

Prove that $\int_0^x \ 1/\sqrt{1-x^2} dx$ is equal to length of unit circle arc?

How to prove that $\int_0^x \ 1/\sqrt{1-x^2} dx$ is equal to length of unit circle arc? I know that the integral is $\arcsin(x)+c$ but really do not see how this is related to arc length.
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1answer
13 views

How can I find the inner limit of a line passing through a lune?

I have a crescent defined by two offset circles with different radii: a small one (let's call it outer circle) centered at (0,0) with radius ...
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1answer
103 views

Characterization of the circle within metric spaces

There are various characterizations of the circle. To be precise, there is not the circle. There are several categories which contain an object we refer to as "the circle". In $\mathsf{Top}$ the ...
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0answers
29 views

Circle Geometry

How do you derive the equation of a circle $(x−a)^2+(y−b)^2=r^2$ if a point on the y-axis is chosen as then you cannot form a triangle and as a result not apply Pythagoras' theorem and derive the ...
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1answer
134 views

Circle construction

I am stuck on this construction: "Show how to construct a circle to pass through two given points and to cut a given circle so that the common chord is of given length". Any clues?
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3answers
58 views

Proving algebraic equations with circle theorems

I got as far as stating that OBP=90˚ (as angle between tangent and radius is always 90˚), and thus CBO=90˚- 2x. CBO=OCB as they are bases in a isosceles. COB=180-90-2x-90-2x. But after this, i am ...
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0answers
39 views

Probability density function for distance between two points.

Two points are chosen randomly inside a circle (and even on the circumference) with radius $r$ What is the probability density function of the distance between the points? I would be very grateful.
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2answers
32 views

Highschool Geometry: Finding Common Tangent

I can't seem to identify what the arrows are indicating in this question, obviously the two lines are parallel but what does it mean? I don't know where to begin. Any suggestions?
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1answer
59 views

Intersection of two circular arcs with same center [closed]

How do you programmatically get intersection points of 2 circles given the same centers, radii, and sweep angle? The 2 circles are not exactly one whole circle. I have an equation for each circle: ...
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2answers
38 views

Calculating if a point is within the overlap of two circles

Two circles of equal radius (R) intersect as shown below. Assuming more points are uniformly distributed in an area with dimensions D*D, where D = 4*R. What is the probability that a point will be ...
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1answer
18 views

Terminals and co-terminals for angles

I'm trying to understand how my teacher converted these angles. I'm not sure if my title is correct but I'm assuming that's what he was doing. For a unit circle he had, \begin{align*} u & = ...
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0answers
31 views

Topological entropy of circle homeomorphism is zero. True or false?

may I know if it is true that $\ f: S^1 \to S^1$ a homeomorphism, then $h_{top}(f) = 0$, where $h_{top}$ stands for topological entropy. I believe this statement is true, but I cannot prove it.
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1answer
106 views

Algorithm - Circle Overlapping

Say you have a shape you want to fill up with circles, where by the circles overlap just enough to cover the whole surface area of the shape. The circles will remain as a fixed size however the shape ...
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0answers
14 views

How do we sketch the ellipse determined by $T(\vec{x})$ and determine its axes, given an expansion factor?

I have been told that if $\left\{\exists \, T(\vec{x})^{-1}\mid T(\vec{x})=A\vec{x} \mid \mathbb{R}^2\mapsto\mathbb{R^2}\right\}$, then the image $T(\Omega)$ of the unit circle $\Omega$ is an ellipse. ...