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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How to prove this theorem rhetorically?

It is not possible for a part of any of three conic sections to be an arc of a circle. It is asked to prove this theorem without using any notation or any modern form of symbolism whatsoever ...
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1answer
45 views

Smallest-circle problem, but with circles instead of points?

I have a growing set of circles (and locations for those circles), each step I add one. I also need the smallest circle that contains all of the circles in my set. I found the wikipedia page about the ...
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1answer
34 views

Given a tangent

EDIT: Solved by sidneyc on reddit and it looks pretty neat: solution Please see this figure of a circle and tangent Given: points A(xA,yA) and B(xB,yB) on a tangent to a circle in point A point ...
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1answer
17 views

What is the easiest way to find the radius and center of the circle of intersection between two spheres?

If given two spheres $S_1$ and $S_2$, of radius $r_1$ and $r_2$, centered at 3-space points $P_1$ and $P_2$, respectively. What is the easiest way to find the radius and center of the circle of ...
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1answer
28 views

Locus of the point of intersection of the pair of perpendicular tangents to the circles $x^2+y^2=1$ and $x^2+y^2=7$

Locus of the point of intersection of the pair of perpendicular tangents to the circles $x^2+y^2=1$ and $x^2+y^2=7$ is the director circle of the circle with radius ...
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1answer
15 views

Followup question about product of slopes on unit circle at rational points

This is a followup question to: Product of slopes of rational points on the unit circle (related to pythagorean triples) mathlove correctly showed that $D=1$ gives an infinity of solution pairs. But ...
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1answer
32 views

Product of slopes of rational points on the unit circle (related to pythagorean triples)

The slope of the line from the origin through a rational point on the unit circle can be written as $$s = \frac{2t}{1-t^2}$$ where t is a rational parameter and $-1<t<1$ I conjecture that for ...
2
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3answers
58 views

Find the angle between two chords passing through points where lines are tangent to the circle

In the given figure, PQ and PR are tangents to the circle with centre O and S is a point on the circle such that $\angle$SQL=50$^{\circ}$ and $\angle$SRM=60$^{\circ}$. Find $\angle$QSR. What I've ...
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1answer
15 views

Find circle from normal vector and second point

I'm given two points on a circle: a point $(x_1, y_1)$ with corresponding normal vector $(u_1, v_1)$ and a second point $(x_2, y_2)$ (without normal vector). How can I compute the circle? ...
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0answers
21 views

Find the Radical Axis of the Circumcircle of Triangle ABC and its Nine Point Circle

Given a triangle ABC, find the radical axis of its circumcircle and its nine point circle.
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2answers
43 views

Circular Arc Prametrization not Using Radius

In an optimization problem I have to parametrize a circular arc. Thus far, I have reduced a more general problem to the figure below: The figure shows a symmetrical circular arc, with chord length ...
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0answers
62 views

Fitting a circle in between three others?

I have three touching circles, and I was trying to find the point inbetween them that is equally far from all three circles. So I created a system of equations: $$ \begin{array}{lcl} (x-x1)^2 + ...
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1answer
32 views

Calculating the central angle of a circle

Given a circle with center $(a,b)$ and radius $r$, oriented counter-clockwise, and two points that sit along the circle, $(x_1,y_1)$ and $(x_2,y_2)$, what the is the great circle distance (GCD) ...
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1answer
39 views

Distance between points at circles

May be someone can help me to solve the problem. There are circle with radius R1 and circle with radius R2. We also know the distance between A and O and that angle AOB = $\phi$. The aim is to ...
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1answer
33 views

Green's Function for a Semi-Circle

Find Green's function for $\Omega = \{(x,y) \in \mathbb{R}^2\mid x^2+y^2 < r^2, y>0 \} $. I tried to find it and I know that $G(x,y)=\frac{1}{2}\pi\cdot \ln\left(\frac{1}{|y-x|}\right) + ...
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0answers
15 views

Does Elzinga & Hearn algorithm depend on initial points

Elzinga & Hearn is an algorithm which find the smallest enclosing circle of $n$ points in plane. I wonder is it a good idea to initialize the algorithm of Elzinga & Hearn with the two ...
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1answer
63 views

Quarter rolled around a quarter.

This question is inspired by this YouTube video which asks a question about how many times a circle revolves when rolled around another circle and comes up with a counterintuitive answer. I have a ...
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1answer
34 views

Place a circle 'on top' of two other circles?

I have two circles (their radii and position) given. I then have a third circle (only it's radius), and would like to calculate its position so it touches both other circles: There are always two ...
0
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1answer
7 views

Regular scale points of a circle, but keeping direction, part 2: the normals

Months ago I had a problem and fortunately, MvG was able to help me: Scale position points in a circle.Look like normal scaling Now I have an additional problem to that problem. I would like to keep ...
7
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1answer
82 views

Question based on chords of a circle

Question: Given a circle and two points $P$ and $Q$ not neccessarily on that circle. Perpendiculars are drawn from points $P$ and $Q$ to the polar lines of the points $Q$ and $P$ respectively. Prove ...
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1answer
32 views

Eccentered circles - determine space between 2 circles at any point around circumference of inner circle

Several months ago I asked the question referenced here .... Eccentered Circles - determine space between circle at a given location For this same question I now need to figure out a more generic ...
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0answers
15 views

How to integrate a distribution in spherical coordinates over a circle?

I have an angular distribution $\frac{s \sigma}{d\Omega} = \frac{d\sigma}{d(cos\theta)d\phi}$. How can I calculate it over a circle which lies on the plane $X = dist$, has radius $r$ and its centre is ...
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0answers
30 views

Derivation of Equation of common chord of two circles(S1-S2) in Cartesion form

I come to know the equation of common chord of circle and it helped me a lot in finding common tangents(when circles touch each other). I want to know it's derivation and I seek help for this!
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2answers
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Double integral with Polar coordinates - hard example

Calculate using polar coordinates: $$\iint_{D}^{} (x^2+y^2)^\frac{1}5 \ dx \ dy $$ where D is the region inside the circle with radius 1. Working: D: $ \ x^2+y^2=1 \\ $ so $ 0 \leq r \leq 1 \ \ , $ ...
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2answers
56 views

Geometry: System of Circles

Given a circle $\,x^2+y^2+dx+ey+c=0,\,$ find the general equation of a circle passing through the intersection of this circle and the line $\,lx+my+n=0.$ My approach was to consider a circle of the ...
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3answers
74 views

Find the center of a circle on a sphere given 2 points and its radius [closed]

How can I find the set of the centers of the circles on sphere that pass through 2 given points and have pretedermined radius, using spherical coordinates? Assume that the radius of sphere is $R$. ...
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0answers
33 views

Evaluating the curve (line) integral of a complex function along three circles in $\mathbb{C}$

I want to find the the curve integral $$\int_γ \frac{1}{1 - z + z^2 - z^3 } dz$$ with $γ$ passing through the following sets counter-clockwise once. a) $\{z \in \mathbb{C}, |z - i| = 1\}$ b) ...
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1answer
49 views

Proving given curve is a circle

Consider a curve $ax^2 + 2hxy + by^2 = 1$ and a point P not on the curve. A line is drawn from the point P which intersects the curve at points $Q$ and $R$. If the product $PQ\cdot PR$ is independent ...
0
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2answers
18 views

Intersection beween circle and line

So I have the circl $x^2+y^2=5$ and the line $y=2x+c$ and I want to find all the points were the two intersect. I know how to solve this for a specific value of $c$, but I do not know how to handle ...
0
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1answer
43 views

Finding Locus of mid point of portion of tangents

Finding locus of middle points of tangents to the circle: $ x^2 + y^2 = a ^2$ terminated by the coordinate axis. I am not able to figure out what the question wanna say... any help is appreciated.
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0answers
13 views

Mobius Transformations and Circular Arcs

Suppose that a region $D$ has as a boundary $\partial D$ which consists of two circular arcs with the same end points (say $a$ and $b$). I want to show that the following Mobius transformation: $w = ...
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2answers
63 views

find the area of the equilateral triangle inscribed in a circle $x^2+y^2+2gx+2fy+c=0$ ?

I am stuck on the following problem: How can I find the area of the equilateral triangle inscribed in a circle $x^2+y^2+2gx+2fy+c=0$ ? Answer is given to be : $\frac{3\sqrt 3}{4}(g^2+f^2-c)$ ...
2
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1answer
30 views

Covering a rectangle with circles

On a rectangle table with area A, n unit-radius circles are placed and it is not possible to place any extra circles without overlapping with some of the existing ones or without placing circle's ...
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0answers
31 views

Finding locus of circle passing through extremities of the two rods

Two thin rods AB and CD of length 2a and 2b moves along OX and OY where O is the origin. Find the locus of the center of the circle passing through the extremities of the two rods. My attempt:- ...
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2answers
49 views

Find the center and radius of the circle whose equation is $x^ 2 + y^ 2 - 6x - 2y + 4 = 0$ [closed]

Find the center and radius of the circle whose equation is $$x^2 + y^2 - 6x - 2y + 4 = 0$$
7
votes
2answers
81 views

Given two intersecting circles finding the coordinates of intersection of common tangent?

Q) A circle $C_{1}$ is drawn having point P on x-axis as its centre and passing through the centre of the circle $C:x^2 +y^2=1$. A common tangent to $C_{1}$ and $C$ touches the circles at Q and ...
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1answer
46 views

Angle of Reflection Inside A Circle

Note: I have browsed and found solutions to problems involving an external collision (angle of reflection along the tangent outside). I am making a program which has a ball inside a larger circle, ...
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1answer
47 views

Ratio of the areas of three concentric circles [closed]

A target is divided into three regions by concentric circles with radii that are in the ratio $1:2:3$ Find the ratio of the areas of the three regions in the form $a:b:c$ where $a$, $b$ and $c$ are ...
2
votes
2answers
29 views

Area of Circles and Sectors

I encountered a question about areas of a circles and sectors on KhanAcademy, I was given the sector area and the central angle of the radian. I know that the ratio of: $${Area\,\,of\,\, Sector \over ...
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1answer
31 views

Finding the center of circle touching a line pair and a point.

A circle passes through the point 3,$\sqrt{\frac{7}{2}}$ and touches the line pair $x^2-y^2-2x+1=0$. The co-ordinates of the centre of the circle are:- My attempt:- Using quadratic formula to ...
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2answers
19 views

Parametric Equation of conics: Parabola

Let $P(ap^2,2ap)$ and $Q(aq^2,2aq)$ be two points on the parabola $y^2=4ax$ such that PQ is the focal chord. Let $A(at^2,2at)$ and $B(as^2,2as)$ be two other variable points on $y^2=4ax$. a) Show ...
2
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3answers
49 views

Proof related to circle

How can I prove that if two circles, one entirely inside the other, intersect at a point, then that point of intersection must be collinear with the centers of the two circles?
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3answers
61 views

find equation of the middle circle

The diagram below is of 3 circles have 3 centres A, B and C and they are collinear. The equations of the circumferences of the outer circles are ${(x + 12)^2 + (y + 15)^2 = 25}$ and ${(x - ...
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1answer
77 views

Maximum number of pythagorean triples on a circle not centered on the origin

Suppose we two equations $$x^2+y^2=r^2$$ and $$(x-a)^2+(y-b)^2=2g^2$$ Where x,y and r are integer variables greater than 0. a,b and g are integer constants greater than 0. I conjecture that for any ...
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2answers
41 views

Finding points in circles

so I have two questions Im stuck on and I really do not know what to do at all. Thank you. 1) 2)
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2answers
17 views

Find the equation of the tangents given the gradient

We're given that $x^2 + (y+2)^2 = 4$ and we're asked to find the equation of the lines where the gradient $=1$ Through implicit differentiation I got $x + (y+2)y'=0$ and if $y'=1$ then: $y=-x-2$ is ...
0
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1answer
34 views

Equation of circle in 3d Plane?

Suppose I have a sphere centered at origin. $$ x^2+y^2+z^2=5 $$ and a plane $$ \vec{r}.(\hat{i}+\hat{j}+\hat{k})=3\sqrt{3} $$ And this plane cuts the sphere at a circular region. How do I write the ...
0
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1answer
32 views

Finding the length of the line segment $JI$

In the diagram, $J$ is the circumcenter of $\Delta ABC$ and $I$ is the midpoint of $BC$. How can i show that $JI=\cfrac {R}{ \cos A}$ ? I simple don't know how to prove such simple fact...
3
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1answer
54 views

Area at centre of Venn circles

How much information do we need to calculate the area of the centre of $3$ Venn circles? I would guess we need to know the lengths of the sides of the triangle formed by the circle centres and the ...
3
votes
2answers
94 views

How to find the area of a square inside a semicircle using only the radius?

Provided with only the radius of the semicircle (10 cm) and the knowledge that the corners of the square touch the semicircle, how can one find the area of this square?