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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Unit circle - how to prevent backward rotation

Let's assume we have a unit circle (0, 2$\pi$). Basically I have a point on this circle who is supposed to move only forward. This point is controlled by the user mouse and constantly calculate 25 ...
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2answers
105 views

Why $\pi r$ is not equal to $2r$?

If there is infinity number of small arcs on top of diameter (can assume it is a simple line which has a length of $2r$) of a half circle (radius is “$r$”) why $\pi r$ is not equal to $2r$?
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2answers
2k views

A unique circle with 3 points proof

I have prove the theorem: There is only one circle passing through three given non-collinear points in both geometrical and algebraic ways. THere is one question that I just have no idea with. 'the ...
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2answers
98 views

Does this alternating sum of roots converge to $\sqrt2$?

This problem arose from what I'm hesitant to call an investigation into a certain type of "quadrature". Starting with the unit disk, I partition it into $p$ pieces by cutting the disk with vertical ...
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2answers
486 views

Pdf for distance between two uniform random points in a circle

This is my first post in the group and I would be very thankful for any help. I am trying to develop a probability distribution for a performance analysis in my thesis. I am trying to look in to ...
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3answers
55 views

Show that the circle drawn on a focal chord of a parabola $y^2=4ax$, as a diameter touches the directrix

Question: Show that the circle drawn on a focal chord of a parabola $y^2=4ax$, as a diameter touches the directrix. Let the parabola be $y^2=4ax$ Let the focal chord be $y = m(x-a) $ Subbing ...
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3answers
36 views

Finding the locus of a point P if the tangents drawn from P to circle x^2 + y^2 = a^2 so that the tangents are perpendicular to each other?

I tried solving this and then I got to this condition here, after I applied the formulua for finding the angle between the tangents Formula is Angle btw tangents = cos(theta) = (1 - tan^2(theta)/2)/ ...
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3answers
360 views

Circle Geometry Questions

In rectangle $ABCD$, we have $AD = 3$ and $AB = 4$. Let $M$ be the midpoint of $\overline{AB}$, and let $X$ be the point such that $MD = MX$, $\angle MDX = 77^\circ$, and $A$ and $X$ lie on opposite ...
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4answers
3k views

Calculate $\pi$ precisely using integrals?

This is probably a very stupid question, but I just learned about integrals so I was wondering what happens if we calculate the integral of $\sqrt{1 - x^2}$ from $-1$ to $1$. We would get the surface ...
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3answers
22 views

Given a tangent a radius, how can you calculate the center point of a circle? [on hold]

How can you calculate the center point (2D) of a circle knowing its radius and a tangent $y = mx + t$?
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1answer
592 views

Best fitting circle to points in 3D

I have a set of n ≥ 3 points in 3D that are measurements of a possible circle. The measured points are "noisy" so best-fitting algorithms are involved. I'm programming in C# and have put together some ...
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1answer
39 views

In the figure, two circles intersect at $P$ and $Q$…

In the figure, two circles intersect at $P $ and $Q$. $O$ is the centre of the smaller circle which lies on the circumference of the larger circle and $RO$ is joined and produced to meet $QS$ at $X$. ...
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0answers
24 views

Sum of Area of Circles. [duplicate]

A circle of radius x cm is inscribed in an equilateral triangle and is tangent at three points. Three smaller circles are inscribed so that they are each tangent to two sides of the triangle and to ...
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1answer
23 views

Dissecting a circle with an irregular rectangular grid

Can a circular disc be 'dissected' by a rectangular grid into a finite number of pieces in such a way that the individual pieces of the circle can be grouped into regions of equal area? Clearly ...
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6answers
47 views

Choosing a value so a line and circle intersect a one, two, and no points

Let l be a line and C be a circle. $y=x+d$, where $d$ is to be determined. $C=x^2+y^2=4$ Pick a value for $d$ so that l and C intersect at one point. Pick a value for $d$ so that l and C ...
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1answer
45 views

Find the radius of three identical circles which touch each other externally.

Three identical circles touch each other externally. The tangents at their point of contact meet at a point whose distance from any point of contact is 2 cm. The radius of the circles is?
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1answer
42 views

Inscribed Shape on Circle given Specific Edges

How would you find the vertices (corners' position) of a shape that inscribes a circle of adjustable radius, given a set of edges? Angles of polygon are not fixed, but edges are. A few examples: ...
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1answer
36 views

Find sides of isosceles triangle inside a circle with line segment lengths as 5 and 4 as shown in the link. pls help!

Pls see the diagram below. I tried to use similar triangles and came to my wits end. Any help will be appreciated!
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1answer
37 views

Problem on circles, tangents and triangles

Let $c_1,c_2,c_3$ be three circles of unit radius touching each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed ...
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1answer
29 views

Formula to map any given point on circumference of circle with given radius

I am working on a project where I need this. Mathematically : I need a formula to map any given point P(x,y) on circumference of a circle of given radius r and center c in 2D space. Insights of ...
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3answers
30 views

Prove that $AH^2+BC^2=4AO^2$

Prove that $AH^2+BC^2=4AO^2$, where $O$ is the circumcentre and $H$ is the orthocentre of the triangle $ABC$.
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38 views

What information is needed to determine a unique circle? [closed]

What information is needed to determine a unique circle? I've been trying to find the answer to this question, but I keep getting results about 3-point proofs, but that is not what I am looking for.
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1answer
24 views

Find the equation of a circle passing through $(-2,4)$ and through the point of intersection of a circle$\dots$ [closed]

Find the equation of a circle passing through $(-2,4)$ and through the point of intersection of a circle $x^2 + y^2 - 2x - 6y + 6 = 0$ and a line $3x+2y-5=0$
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1answer
47 views

Largest enclosed (inscribed) circle in cloud of points

I have a set of points that approximately lie on a circle. I would like to compute the largest circle that does not contain any of the points. Of course, one could draw the circle far away from the ...
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2answers
47 views

Point A is picked randomly in a circle with a radius of 1, and center O. What is the variance of length OA?

Point A is picked randomly in a circle with a radius of 1, and center O. What is the variance of length OA? I believe the CDF has to found first, then we need differentiate it, find the expected ...
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2answers
54 views

How to find the tangency condition for this circle geometry problem?

Suppose I have a circle $C$ of radius $1$, and I have a chord of this circle, of given length $l$. The chord makes a known angle $\theta$ with the tangent to the circle. I position a smaller circle $...
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1answer
16 views

Find the measurement of…

Find the measurement of unknown angle in the given circle with centre at $O$. My Attempt $1$. $\angle QRS=\angle PQR$ $2$. $\angle PQR=\angle PMR$. I am struck here. Please help me with this ...
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2answers
48 views

Finding Perimeter of Shape

"Two circles of radii 5cm and 12cm overlap so that the distance between their centers is 13cm. Find the perimeter of the shape." This question was from a chapter about circle measure under the length ...
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1answer
20 views

how to get the angle of arc ??

dart game board is divided into sectors by 30 degrees like pizza slice. the given is (x, y) coordinates, and I need to find where coordinates are lying on. how can I get the angle just with ...
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0answers
10 views

Probability Distribution over successive circular arcs.

So I'm looking at a problem sketched out below: so here what happens is you have a particle which moves at a constant speed and has some current orientation. At each timestep it can change it's ...
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4answers
129 views

Common tangents to circle $x^2+y^2=\frac{1}{2}$ and parabola $y^2=4x$

I'm having trouble with this. What i do is say $\epsilon: y=mx+b$ is the tangent and it meets the circle at $M_1(x_1,y_1)$, i equate the $y$ of the tangent with the circle: $y=\pm \sqrt{1/2-x^2}$ and ...
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1answer
19 views

Formula for calculating x2 and y2 of a line that behaves like a clock hand?

In the image below, the diameter of the circle is 100, and x1=50 and y1=50 for the line's starting point in the middle. I'd like to be able to draw the line so that it is pointing at different ...
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1answer
46 views

Given: 2 lines containing the diameter of a circle and a point lying on this circle; Find: the equation of this circle

The lines $ y = \frac{4}{3}x - \frac{5}{3} $ and $ y = \frac{-4}{3}x - \frac{13}{3} $ each contain diameters of a circle. and the point $ (-5, 0) $ is also on that circle. Find the equation of this ...
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2answers
521 views

Intersection of a point and absolute value function contained within a circle

I'm attempting some crazy ideas while programming a game and ran into the following math problem that has been bugging me for a few days: Given a unit circle and a random point $P$ within the circle, ...
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2answers
44 views

Area of a circle on sphere

On a (flat) Euclidean plane, the area of a circle with a radius $r$ can be described by the function $A(r) = \pi r^2.$ But how can one describe the area of the same circle on a spherical manifold? ...
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2answers
133 views

Finding the radius of the smallest circle that can circumscribe an equilateral triangle

Q:A puzzle board is in the form of an equilateral triangle that has an area of $7\sqrt{3}$ if the board is placed on a circular table, what should be the min area of the table so that the whole board ...
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1answer
408 views

max points in circle given radius and min spacing between points

I want to know how many points ($n$) can be placed in a circle of radius $r$, with a minimum spacing $s$ between points. I find postings for several similar problems -- smallest circle around a set of ...
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2answers
34 views

How to find the equation for the circle when…

A circle goes trough two points, $A=(-1,2)$ and $B=(3,0$). You also know that the center of the circle is an element of the following linear equation: $$k \leftrightarrow 2x+y+3=0 .$$ How can you ...
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1answer
33 views

Find radius of circle (or sphere) given segment area (or cap volume) and chord length

The goal is to design a container (partial sphere) of given volume which attached to a plane via a port of a given radius. I believe this can be done as follows but the calculation is causing me ...
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2answers
57 views

If the circles $x^2+y^2…$

If the circles $x^2+y^2+2ax+c^2=0$ and $x^2+y^2+2by+c^2=0$ touch externally, prove that $\frac {1}{a^2} +\frac {1}{b^2}=\frac {1}{c^2}$. My Attempt Here $$x^2+y^2+2ax+c^2=0$$ $$x^2+2.x.a+a^2-a^2+...
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1answer
516 views

Computing a matrix to convert an (x,y) point on an ellipse to a circle

I have an ellipse defined by its semi-major axis, inclination, and position angle. The ellipse is centered on the origin. I would like to solve for a matrix that converts this ellipse to a circle. ...
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1answer
71 views

Locus of the center of the circle of radius $a$,which always intersects coordinate axes

If the axes are rectangular, show that the locus of the center of the circle of radius $a$,which always intersects coordinate axes is $x\sqrt{a^2-y^2-z^2}+y\sqrt{a^2-z^2-x^2}+z\sqrt{a^2-x^2-y^2}=a^2$ ...
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2answers
38 views

Find the equation of a circle…

Find the equation of a circle with radius 4 units, whose Centre lies on the line $4x+13y=32$ and which touches the line $4x+3y+28=0$. I could only make a figure with the help of the question. can ...
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2answers
452 views

Find the maximum perpendicular height between a chord and an arc.

I am doing a maths modelling project, and I am stuck on a part. I have a (arc length) and L (chord length), but I want to find H, the maximum perpendicular distance between them! Any help would be ...
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2answers
21 views

Finding Radius Of Circle From Circle's Equation

For basic equations like:- $$ x^2 + y^2 = 4 $$ we can find out that the radius of the circle is 2. But for an equation like:- $$ x^2 + (y+1)^2 = 1 $$ What will be the radius of the circle?
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Cutting a pie into 2 unequal peices with a single cut, minimising its length. [closed]

Suppose we have a circle with an area of 1, which we are to cut into two pieces, of area (x) and (1-x) respectively. Let x<0.5. How should we make the cut, to minimise its length? What is the ...
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0answers
41 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
555 views

Inner tangent between two circles formula

As a programmer I need to draw the inner tangents between two circles, but only the segments, not the whole line. But the internet is surprisingly hostile to lazy programmers who don't know their ...
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5answers
1k views

How to prove the infinite number of sides in a circle?

I was in geometry class today when I came across the following formula for the external angle of a regular polygon with n sides: $$Ea = \frac{360º}{n}$$ So I thought if $$ n\rightarrow\infty $$ then $...