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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1answer
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Largest equation of a circle that shares 2 tangents with a curve

Just played around on a graphic calculator a little, and discovered that given the curve $y=x^2$ , all circles with the equations in the form of $\left(y-a\right)^2+x^2=\frac{4a-1}{4}$ for all $a>0....
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4answers
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Find the center of the circle through the points $(-1,0,0),(0,2,0),(0,0,3).$

Find the center of the circle through the points $(-1,0,0),(0,2,0),(0,0,3).$ Let the circle passes through the sphere $x^2+y^2+z^2+2ux+2vy+2wz+d=0$ and the plane $Ax+By+Cz+D=0$ So the equation of ...
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0answers
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An geometry problem. proved that the circles inscribed in triangle ABD&CAD are tough each other.

The inner circle of triangle $ABC$ touches $BC$ at $D$ . Show that the circles inscribed in triangles $ABD$ and $CAD$ touch each other.
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1answer
53 views

Geometric question consisting of a rectangle and two circles

Let ABCD be a rectangle. let P be a point on the 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 and R. Prove that ...
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1answer
43 views

equilateral triangle and inscribed circle

Let ABC be an equilateral triangle. let D be a random point on BC. Let I_1 and I_2 be the incenters in ABD and ADC. Let O be the circumcentre of AI_1I_2. Prove that OD is perpendicular to BC.
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2answers
52 views

Equation of a tangent on a circle given the gradient and equation of the circle

My maths teacher told me this problem was impossible without knowledge of implicit differentiation: is she right? You are given the equation of the circle $\left(x+2\right)^2+\left(y-2\right)^2=16$ , ...
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1answer
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Geometry problem that I can't solve (polygons inscribed in a circle).

I encountered this problem earlier and couldn't figure out how to solve it. I couldn't get all that far with the problem. The red marks were added by me, couldn't figure out anything other than those ...
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3answers
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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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3answers
37 views

How to solve the word problem below?

Can anyone guide me through this problem? I know how to solve the equation of the circle (the Earth) below but I don't know how to solve the equation of the orbit.
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1answer
29 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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1answer
25 views

Given a rectangle and some intersecting circles, are these points collinear?

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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2answers
36 views

Equation of Circle

Prove that the equation $x^2+y^2+2gx+2fy+c=0$ always represents a circle. I just don't have any idea regarding this. Can anyone help me? Help much appreciated! Thanks..
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0answers
62 views

How to decompose a 2d shape into sin and cosin modes?

Assume that you have a circle with radius $r_0$, then you keep adding cosine modes as below: $r=r_0+a_1\cos(1\theta)+a_2\cos(2\theta)+a_3\cos(3\theta)+a_4\cos(4\theta)+~...$ if you plot this as ...
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1answer
34 views

Circle Tangent problem

Triangle $ABC$ is isosceles with $AB=AC$. A circle that is tangent to line $AB$ at $B$ intersects line $AC$ at points $P$ and $Q$. Prove that $BC$ bisects angle $ \widehat{PBQ}$.
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1answer
64 views

What is the average distance between two random points inside a circle?

Assume you have a circle with some radius r. What is the average distance between two random points inside the circle? (Edit: This is different from this already answered question, because here the ...
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3answers
27 views

Finding a surface on which a given curve lies

A curve is parametrized $r(t)=(cost,sint-1,2-2sint), \quad 0\le t\le2\pi$ Find three different surfaces on which C lies. I have managed to find two surfaces visually: $2y+z=0$ and $x^2+(y+1)^2=1$ ...
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0answers
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How do I derive this correctly? [closed]

Given 70 meters of material, which would be the values of r and h to give the biggest area possible. How do I derive this formula? This is where h and r come from. A being Area P being perimeter $$...
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2answers
29 views

UIL Math Contest Problem #60 Regional

Given the circle O with perpendicular diameters and a chord, find the area of the circle if EF = 8" and DE = 20" (DF = 12"). (integer) DE is a chord that intersects one of the diameters and shares a ...
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1answer
43 views

Analytical expression for varying center of mass

Imagine having a circle with center of mass in orego. I need stepwise to remove a strip from that circle (i.e. removing an arc of the circle), and for each step locate the center of mass. An ...
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1answer
37 views

Make 3 circles intersect in only one point by changing their radius as little as possible

I have 3 circles in a 2D space with known center coordinates ((xa,ya),(xb,yb),(xc,yc)). The circles don't intersect at a unique point but they should be close to it. I would like to find the point (...
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1answer
28 views

For a ring of n tangent circles inscribed within the perimeter of a larger circle, calculate radius or diameter of circles or n

Update: K. Jiang solved for n, so I updated my question here below to now have all of the working formulas for easy reference to others. Original Question: I have the following documented below ...
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1answer
27 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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1answer
30 views

two triangles in a circle

If the angles$A=B=10$ and the bow $AM=40$ then find the measure of bow $BN$.($O$ is the center. My Attempt:The angle $AOM$ equal to $40$ degrees then the angle $AON$ equal to $140$ degrees then ...
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1answer
21 views

Angle of circular droplet

I am trying to find the angle $\theta$ of the following droplet: I think using $\tan$ is the right way to go, and I thought of using it on the angle formed by the line $r$ and $b$. However, that ...
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1answer
20 views

Analytical Expression for a radius in the circle geometry figure shown

I want to find an expression for r in terms of the other parameters mentioned. This algorithm will then be used inside an microcontroller , to control the existing machinery Best Regards TaimoorAli
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1answer
25 views

Circle Geometry - How to find Central Angle

I am currently having issue with this Circle Geometry Problem: As noted in the picture, I know that the two triangle are similar. However, I am unable to find the other two angles. Thanks for any ...
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1answer
24 views

Circle Geometry - What Information Given can help me find the other angles. [closed]

I'm currently studying for a test in Grade 8 Honours Math (Grade 9). We have some questions involving Circle Geometry, but I'm having trouble here: You see, I know the value of w due to it being an ...
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0answers
29 views

How so I work out how many circles per ring / which ring a circle is on

Lets say I have a bunch of circles I want to arrange in rings. The circles start in the centre and move to a new ring with a bigger diameter each time an inner ring runs out of room. How would I know ...
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2answers
36 views

three symmetrically placed equal circles

If three symmetrically placed equal circles intersect in a single point, as illustrated in the figure. What should be the distance between the centre of the three circle for obtaining such a overlap.
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2answers
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calculate the correct space between dots on a dashed circle, to have a perfect alignment

I want to draw a dashed circle, with a diameter D, and X dots composing the circle, like on this image: dashed circle How can I define the exact space I should have between the different dots, to ...
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1answer
83 views

$\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ - Theory of distribution

An exercise asks to find the wave trace $w(t)=\operatorname{tr} \left(e^{it \sqrt\Delta}\right)=\sum_{k \geq 1} e^{it \sqrt{\lambda_k}}$ as a distribution (or generalized function) of the Laplacian ...
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5answers
53 views

Finding a Coordinate on a circle using radius, angle, and origin

I am trying to calculate a point on a circle using an angle and a different point. With this picture, I know the origin O, the radius r, the angle A, and the point B. Now I want to find the point C....
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2answers
58 views

A triangle in a circle

According to the following picture $E$ is the midpoint of $BD$ and $DC=BD$. If measure of $\angle EGF$ is equals to $90$ degrees then find the value of $\frac {DE} {EF}$.(point A is the center and BC ...
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1answer
48 views

Circle tangent to three tangent circles (without the Soddy/Descartes formula)

We have three circles tangent to each other with radii $1$, $2$, and $3$. Another circle is tangent to the other circles; find the radius of that circle using elementary geometry, without the Soddy ...
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2answers
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Need help with alternative method to equation of a tangent at the point of a circle

so I know a simpler of looking for the equation of a tangent at the point of a circle is to differentiate, my lecturer would rather we not use calculus and has charged us with looking for an alternate ...
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2answers
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Why does convention not recognize $2\pi$ as the fundamental quantity? [duplicate]

It looks as though my question might turn out to be a duplicate. If so, it does not need an answer, after all, thanks. ORIGINAL QUESTION Why does convention recognize $\pi\approx 3.14$, rather than $...
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3answers
38 views

Process for solving this system of equations

I have this system of equations for which I'd like to solve for $x$,$y$, and $r$ where $a$,$b$, and $t$ are constants: 1: $0 = (x-a)^2 + (y - b)^2 - t^2$ 2: $y = \dfrac{bx-rx+ar}{a}$ 3: $r = \dfrac{...
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1answer
20 views

The Locus Of M (Repeated Questuon) [duplicate]

Let A and B be two fixed points on a straight line. Two circles touch this line at A and B respectively and the tangent to each other at M, when the circles vary the locus of M is? This question has ...
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0answers
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Increasing or decreasing theta based on direction of vector

I am a programmer, and I have some holes in my math knowledge that I am working on filling in. Right now I'm working with a simple process involving drawing curves and straight lines. The line that is ...
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2answers
37 views

Show that there are at most two rational points on $(x - a)^2 + (y - b)^2 = r^2$ for $a, b$ irrational.

For any given irrational numbers $a, b$ and real number $r \gt 0$, show that there are at most two rational points (points whose coordinates are both rational numbers) on the circle $(x - a)^2 + (y - ...
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2answers
35 views

Converting the Great Circle distance to direct distance between two points on earth?

Apologies if this question has been asked before. Across the surface of the Earth, the distance between London and New York is 5567 km. Given that the earth has a radius of 6371 km, what is the ...
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3answers
49 views

Prove that : A circle consist of infinite points

How to prove a circle consist of infinite points ?Proof using calculas or computational theory is appreciated?
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2answers
55 views

Find nearest points on the circumference of a circle based on reference coordinates and centre coordinates given

For a programming purpose I've been asked to plot few points next to a point on a circular diagram. The only given values are the reference point coordinates and distance from/to of the new point to ...
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0answers
42 views

Volume of “the complex projective space” of a certain radius.

Consider the circle action on $\mathbb C^n$ given by $(e^{it},z)\to e^{it}z$. A moment map for this action is $J:\mathbb C^n\to\mathbb R:z\to -\frac{1}{2}|z|^2$. Let $M_l=J^{-1}(-\frac{l}{2})/U(1)$ ...
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2answers
126 views

Wave kernel for the circle $\mathbb{S}^1$ - Poisson Summation Formula

Question : How could I compute the (wave) kernel from the fact I have already found (wave) trace on unit circle? The definitions are related to the page $25$ of the following pdf. As the Spectrum$(S^...
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2answers
110 views

Minimum Area of An Ellipse Surrounding Four Circles

The circles are all four combinations of $(x\pm60)^2+(y\pm25)^2=5^2$ (see pic at end). The ellipse I've got is one I found via trial and error but there must be an analytical way to solve this, right?...
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1answer
36 views

Cirle's Center and Radius for Lots of Point

I know that If I have 3 points I will have this center (I calculated this) a=\left[\begin{matrix}x1^2+y1^2&y1&1\\x2^2+y2^2&y2&1\\x3^2+y3^2&y3&1\\\end{matrix}/ 2*\left[\begin{...
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1answer
31 views

Geometry including semicircle and arc length

This is a question in the Princeton online test of GRE general book. I got it wrong, but even when I looked at the answer I find it difficult to understand how the answer is obtained. In the ...
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1answer
58 views

Sine law and circumscribed circle

How is $\frac{a}{\sin(A)}=2R$ (where $R$ is the radius) derived?
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2answers
40 views

Putting Numbers on a Circle

If I have a circle and I start numbering points along the circumference with all the natural numbers: 1, 2, 3, 4, and so on, such that the length of the arc between two consecutive numbers is constant,...