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

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1answer
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Prove that the triangle areas are proportional to the radii

The line $MN$ is the radical axis I created. Because of its properties, we have $EM=MF, HN=NG, IQ=QL$, and it is perpendicular to $AC$. Everything is as you see on the diagram below. Here $(ABC)$ ...
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0answers
66 views

What is the 'optimal' equal-area partition of a circle?

What is the (an?) n-partition of a circle that meets the following criteria: The boundaries of each partition can be represented as a union of finitely many finite-piecewise-smooth simple closed ...
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3answers
113 views

Diameter of a circle touching three inner circles of diameter 1

If the diameters of three three inner circle are $1$ meter, what is the radius of the big circle? (Note: the OP provided own answer below, after getting a hint).
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1answer
47 views

Algebraic proof for sphere/circle overlap formula

Two spheres or circles denoted by center position vector and radius $ p_0, r_0$ and $p1, r_1$ will overlap if $$ |p_0-p_1| < r_0+r_1$$ I understand geometrically why it works, but how would one ...
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2answers
116 views

Finding the exact area of a circle?

Background: I recently began taking calculus and it has come to alter the way I look at circles, and curves. The equation of a circle is $\pi r^2$, traditionally in school we have always left the ...
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2answers
22 views

Co Ordinate Geometry Of The Circle

Hi hello this is my last resort as I have no clue how to do these sort of sums and need help badly. Find the equation of the tangent to the given circle at the given point Circle (x - 4)² + (y + 2)² ...
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1answer
132 views

AB is the chord of circle with centre O and DOC is a line segment originating from point

AB is the chord of circle with centre O and DOC is a line segment originating from point D on the circle and intersecting AB produced at C such that BC=OD.IF $\angle BCD=20degree$ then$\angle AOD$?
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2answers
70 views

How can I visually imagine the area of a circle divided by $\pi$?

If I have a circle with an area of 100 units^2, and I divide it by $\pi$, how can I imagine that visually in my mind? Since 100 / $\pi$ =~ 31.83, and the square of that is =~ 5.64, I currently ...
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2answers
51 views

Is that possible that a inscribe angle can be greater than 90 degree

I have found a question like following: Its asked that what could be the angle x if BC is not diameter of the circle. So, my question is if it possible to be greater then 90 for an angle like x? ...
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4answers
211 views

How to find the intersection point of two moving circles?

I'm trying to develop a simulation in C#, and I have to find the intersection (or collision) point of two moving circles, in 2D space. Actually one of my circles will be stationary, so only one of ...
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4answers
81 views

Given circle and point, where does the tangential line through the point touch the circle?

Given a circle with known center $c$, known radius $r$ and perimeter point $x$: $$ (x - c_x)^2 + (y - c_y)^2 = r^2 $$ with a tangent line that also goes through a point $p$ lying outside the circle. ...
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2answers
76 views

Any two points inside a circle are within a diameter of each other.

In many problems involving the Pigeonhole Principle, we often assume the following lemma: Lemma: The distance between any two points in a circle of radius $r$ is at most $2r$. Intuitively, this ...
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0answers
100 views

Equation for Circle in 3D Space Given Center, Radius, and Point

I'm looking for how to derive the equation of a circle, in 3D space, given the following information: The Center Point The Radius One point on the circle I understand that this is functionally the ...
2
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1answer
38 views

Equation of circle - from chord

Question: If one of the diameters of the circle $x^2 + y^2-2x-6y+6 = 0$ is a chord to the circle with center (2, 1), then the radius of the circle is: $\sqrt3,\sqrt2,3,2$ I have no clue as to where ...
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1answer
56 views

Circles and tangents and circumcircles

Question: Tangents drawn from the point $P(1, 8)$ to the circle $x^2 + y^2 -6x -4y -11=0$ touch the circle at the points $A$ and $B$. What is the equation of the circumcircle of the triangle $PAB$? I ...
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3answers
36 views

Compute the set of points (x,y) for a circle of arbitrary radius, with a 1 degree step, without using any trigonometric function.

Is it possible for a computer program to geometrically construct a approximate circle (pixels have line drawing limitations) without using any trigonometric function? e.g. taking the unit circle as ...
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3answers
59 views

How to find the radius of this middle circle arranged as shown.

There is this maths competition geometry problem and my approach. And this is my initial approach. From the picture, the shaded circle looks slightly bigger. What we are looking for is the $x$ ...
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1answer
51 views

Homeomorphism of a Genus-2 Surface

Does there exist a homeomorphism from a genus-2 surface, the connected sum of 2 tori, to two circles, $S^1$, intersecting at a point? Intuitively it seems that the double torus can be squeezed into ...
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2answers
146 views

Midpoint of chord of contact

Question: The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x - 5y = 20$ to the circle $x^2 + y^2 = 9$ is: a) $20(x^2 - y^2)- 36x + 45y = ...
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1answer
66 views

How is circle closed?

I have this thought that circle in 'real' is not a closed figure. We all know that 'pi' is irrational.And integers are nodes in a 'monstrous' line of real numbers. Irrational numbers are ...
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0answers
58 views

Find circles that completely cover a polygon minimizing the amount of space covered outside the polygon

I have an arbitrary polygon that I need to roughly represent using circles. Any point inside the polygon must lie inside a circle. There will be points outside the polygon that will fall under a ...
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2answers
28 views

The length of the side of the square is 4 Find the radius of the smaller circle? [closed]

The length of the side of the square is 4. Find the radius of the smaller circle?
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2answers
28 views

Tangent - point of contact

Question: Tangent to the curve $y = x^2 + 6$ at point P(1, 7) touches the circle $x^2 + y^2 + 16x + 12y + c = 0$ at a point Q. Then the coordinates of Q are: 1) (-6, -11) 2) (-9, -13) 3) (-10, -15) 4) ...
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2answers
46 views

Circle - finding the equation

Question: A circle touches the lines $2x+3y+1=0$ at the point (1, -1) and is orthogonal to the circle which has the line segment having end points (0, -1) and (-2, 3) as the diameter. What is the ...
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3answers
71 views

Solve the length AB (the dashed line)

Can someone show me how I can solve this? (Step by step example with solution appreciated a lot as I am currently practicing). EDIT: After a closer look, it looks as if this is an Isosceles ...
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3answers
82 views

The “Circle” is a Vector Space?

Consider the set of angles $C = [0, \ 2\pi)$ and, for all $x,y \in C$, define the $sum$ operation as the sum modulo $[0, \ 2\pi)$. The identity element of the addition is the angle $0$. The inverse ...
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2answers
40 views

Radius of a curvature

I have a lens (magnifying glass) and I want to calculate the radius of the curvatures on its sides. The lens in question diameter of the lens = 6 cm thickness at center = 7 mm thickness at edge = ...
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1answer
47 views

length of secant line.

I'm looking for way to find the length of a secant line intersecting another line through the center of a circle with a known radius. The intersection point is on the circle and the angle between 2 ...
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2answers
42 views

Circles - tangent from common point

Given the equation of a circle and the points of contact of two tangents, is it possible to find their point of intersection? The obvious method is to find the equation of the two tangents, using the ...
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4answers
76 views

Circles - point of intersection of tangents

Question: Let A be the center of the cricle $x^2 + y^2 - 2x-4y-20=0$. Suppose that the tangents at the points B(1,7) and D(4,-2) on the cricle meet at point C. Find the area of the quadrilateral ABCD. ...
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1answer
59 views

Inscribed Circles in Triangles

This question appeared in this year's UNSW Maths competition. It was question 5b and it was the only question that i couldn't do. Sorry if my explanation is bad as it is complicated to understand ...
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3answers
42 views

Finding the variable of a coordinate point on a circle

This might be a very simple question but I am having trouble figuring it out, so if anyone can explain: A circle is marked with three points A(-3,2),...
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2answers
131 views

Finding the points where a circle intersects an axis

A circle has the equation: x²+y²+4x-2y-11 = 0 What would be the coordinates of the points where the circle intersects with the y-axis and how would you calculate it?
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2answers
51 views

Solving for $\theta$ in a circle

Let's say you have a pendulum hanging straight down and touching the ground at the lowest point. The pendulum has length $l$. If you pull the pendulum back so that the end is height $h$ above the ...
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1answer
36 views

Rules of Inscribed Angles

https://www.dropbox.com/s/chbs2vilr9wjkvz/20140819_130744.jpg Image of question found above. I don't understand why angle BCD is formed by tangent and chord and is equal to 1/2 of arc BC.
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Placing a circle in a square lattice

Two part question. Consider the square lattice $\mathbb{Z}^2$: Imagine you are going to place a circle of radius $r$ somewhere in $\mathbb{R}^2$. Question 1: What is the radius of the largest ...
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1answer
68 views

Enlarge 3 Circles about the same factor to find the Intersect

I currently have 3 circles that not intersect at all. Like this: Now i would like enlarge the circles about the same factor to find the intersection of this three circles. I have tried following ...
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1answer
72 views

Convex optimization approximation

Consider the optimization problem $\mathcal{P}_0$ $$ \min_{x \in \mathbb{R}^2} \left\| x-p \right\|^2 $$ $$ \text{sub. to: } \ A x \leq b, \ \ x_1^2 + x_2^2 = 1 $$ where $p \in \mathbb{R}^2$ is a ...
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50 views

Finding circle which touches two functions

I wasn't sure whether my issue is with my Mathematica code or the actual way I am trying to figure out my problem so if it is a Mathematica issue I can ask it on that stack exchange. Firstly I have ...
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0answers
51 views

What is the curve's name for the “reciprocal” equation of a circle?

The equation of a unit circle is $$(x-a)^2+(y-b)^2=r^2$$ When the origin $$(a, b)=(0,0)$$ the equation becomes $$y=(1-x^2)^{1/2}$$ Naturally when this equation is plotted on graph paper we get a ...
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2answers
57 views

Intersection of circle and ellipse

I'm looking for the points of intersection of a circle $x^2 + y^2 = r^2$ ($r$ is known, origin is $(0,0)$) and an ellipse $(x - x_0)^2 / a^2 + (y-y_0)^2 / b^2 = 1$ ($a,b,x_0,y_0$ are known). ...
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1answer
86 views

Distance to the perimeter of a circle with given radius, distance traveled from origin, and direction

I am programmer by trade but am running into some trouble with a geometry problem. I basically want to start at the center of a circle, travel any distance within the radius, turn any direction, and ...
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1answer
202 views

Find area of shaded region in circle

I am working on this SAT question. Progress AD = 3 largest radius =3 second largest = 2
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2answers
369 views

Circle Line segment intersection

I have a circle with radius r and center $(c_x, c_y)$. I have a line segment $(x_1, y_1)$ and $(x_2, y_2)$ given $(x_2, y_2)$ is always a point inside the circle. I am trying to find the ...
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1answer
245 views

How to find the area of intersection of two circles using axiomatic geometry?

Problem: square(ABCD) is a regular square, and a circle touches internally in the square. Also, arc(BD) divides the square. Then calculate the area of the colored region. This question is easily ...
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1answer
124 views

Not understanding arc midpoint computation

I'm trying to find the midpoint of an arc, so I found this page wherein Gregory V. Akulov and Oleksandr G. Akulov describe the midpoint formula. I pasted the formula & description from the site ...
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4answers
114 views

Area of a circle sector

I have been given the following proportional relationship to derive the area of a circle's sector: $\large\frac{\text{ Area of the sector}}{\text{Area of the circle}}=\frac{\text{arc ...
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1answer
50 views

Work out center of a partial circle

If I have a small section of a circle, inside a square. I know the height and the width of the square and therefore the width and height of the arc, what would be the quickest (not necessarily the ...
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1answer
63 views

Unit circle can't be covered by one chart

I am hoping that someone can give me a proof showing why the unit circle cannot be covered by one coordinate chart, or a reference where I can find a proof.
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1answer
51 views

Can we find a point $M$ on the unit circle such that $\prod_{i=1}^n MA_i=1$?

We are on $\Bbb{R}^2$. Let $A_1,\cdots,A_n$ be $n$ points on the unit circle. Can we find a point $M$ on the unit circle such that $\prod_{i=1}^n MA_i=1$ ? ( of course I mean the distance ...