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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If tangents are drawn from two points which are equidistant from given point, then find the locus

Tangents are drawn to the circle $x^2+y^2=a^2$ from two points on the $X$ axis equidistant from the point $(k,0)$ prove locus of their intersection is $ky​^2=a^2(k-x)$. If I take points as $(k+\alpha,...
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2answers
68 views

If $x^2=\lambda$, then find the value of $\lambda$

A Circle $C_1$ is drawn having any 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 $C$ intersects the circle at ...
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1answer
205 views

Can't understand this solution.

I came across a problem which was already present on the internet. If an arc with a length of $12\pi$ is $\frac{3}{4}$ of the circumference of the circle, what is the shortest distance between ...
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3answers
23 views

Find elevation of a tangent in a circle

I am trying to understand how I can calculate an elevation (i.e. the distance) of a tangent line given an arc and radius. For example : Given that I know $d$ and $s$, how do I get the value for $?$...
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1answer
32 views

Angles of lines tangential to a circle

I am looking to find the angles of line features relative to the tangent of a circle. Please see this example for general idea. Angles to line features (purple) I am looking for are (poorly drawn) ...
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0answers
38 views

Homeomorphisms of the circle

I know that there is a vaste litterature about the group of the homeomorphisms of the circle. I would a good reference to start the study of this topic. Thanks in advance.
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1answer
34 views

Show that three circles are coaxal

Let $A_1, A_2, A_3, A_4$ are collinear, $B_1, B_2, B_3, B_4$ are collinear. Such that $A_1, A_2, B_2, B_1$ lie on circle $(O_1)$, and $A_3, A_4, B_4, B_3$ lie on circle $(O_2)$. Let $MNPQ$ be the ...
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1answer
491 views

Do the tangents of two circles define concentric circles?

Given two non-overlapping circles, $R_1$ and $R_2$. The radii of $R_1$ and $R_2$ may be different. The distance between the centers of $R_1$ and $R_2$ is defined as $x$. Draw the four tangents ...
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2answers
34 views

What would be the area of this Red Marked points? And how to calculate this?

I have been given the length $L$ and the width $W$ of a rectangle and the radius $R$ of circle which is situated in the center of the rectangle . I need to find the area of the red marked portion. ...
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2answers
23 views

Calculating length of vertical line bisecting parallel arcs

I have 2 arcs, offset from one another (never intersecting) and a vertical line through them both (NOT at the center of the arcs). Is there a way to calculate the vertical distance between the 2 arcs? ...
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2answers
34 views

Deriving surface area of a sphere from the circumference

given the circumference of a circle, which is 2πr, how many times do I have to add it to itself to cover a whole surface of a sphere and deriving 4πr^2?
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3answers
60 views

Finding area of a part of a circle

I have the values of $L$, $R$ and $W$ in the picture below. The circle is drawn though the center of the rectangle. And the circle will always intersect the rectangle. How can I find the area of the ...
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1answer
51 views

How to check two circles are linked or not? (without using topology)

In $\mathbb{R}^6$, three loops $$C_1:=\{(0,x,-x;0,y,-y)\mid x^2+y^2=1\}\\ C_2:=\{(x,0,-x;y,0,-y)\mid x^2+y^2=1\}\\ C_3:=\{(x,-x,0;y,-y,0)\mid x^2+y^2=1\}$$ are embedded. Is there a pair of circles ...
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0answers
23 views

A generalization of the Sawayama lemma

Let $ABC$ be a triangle, let $D$ be a point on the line $BC$. The Thebault circle is a circle tangent $AD, BC$ and the circumcircle (yeallow circles in the following figure). I give a ...
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2answers
523 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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1answer
26 views

The intuition behind choosing this length?

In Problem 1, RMO 2004 there is a particular choice of length which leads to the solution, the length being that of the tangent from the foot of the perpendicular to the circle. Just a rundown of ...
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2answers
129 views

how is it that $\int_0^x 2\pi r\ dr$ is equal to the area of a circle [on hold]

I'm studying calculus and I'm having some basic questions, this one is regarding the area of a circle. we know, from some guy, that the circumference of a circle is $2 \pi r$ and the area can be seen ...
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1answer
32 views

Proving circumcenter lies on altitude

Problem: In $\triangle ABC$, let $D$ be the intersection of the tangents to the circumcircle at $B$ and $C$, let $B'$ be the reflection of $B$ across $AC$, let $C'$ be the reflection of $C$ across $AB$...
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2answers
468 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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1answer
528 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
24 views

How to calculate the shortest rotation from current to the target angle? [on hold]

In the following situation: My current angle is 40*, my target angle is 130*. How should I calculate the rotation that should be done to reach the target angle from the current one? I've done the ...
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1answer
38 views

Locus of intersection point of perpendicular tangents

Here is the question which I am referring to:A tangent is drawn to the circle $(x-a)^2+y^2=b^2$ and a perpendicular tangent to the circle $(x+a)^2+y^2=c^2$,find locus of their point of intersection. ...
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1answer
936 views

Calculus Riemann sums for circle and ellipse

I ran into this problem today. I need to compare the Riemann sums for a circle and an ellipse. I have no idea as where to start. Here's the question:
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1answer
429 views

The surface area of a ring: $\pi[(r+dr)^2 - r^2]$ or $2\pi r\,dr$?

I know this may be really simple but here it is nonetheless. Let's say that I have a ring with a radius of $r$ and width of $dr$. I'm trying to find the surface $dS$ of the ring. Isn't it $dS = \pi[(r+...
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1answer
22 views

Why Circle is traced counterclockwise and ellipse is traced clock wise?

In the Lecture 32: Polar Coordinates,professor traces the circle counterclockwise, but traces the ellipse clockwise. "Which was this one here. And first we noted that this does parameterize, as we ...
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3answers
42 views

What happens when $r \to \infty$? Will it be a line? (partial circle)

Let $a$ be a arc of particle circles, which is constant. What happens when $r \to \infty$? Will it be a line? Radius of partial circle : $r$, Arc of partial circle : $a$ and constant, For $r=r_0$ ...
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1answer
48 views

Circle Problem:Which of the following are true

If the circle $x^2+y^2+2gx+2fy+c=0$ cuts the three circles $x^2+y^2−5=0$, $x^2+y^2−8x−6y+10=0$ and $x^2+y^2−4x+2y−2=0$ at the extremities of their diameters, then which of the following are true ? $c=...
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1answer
399 views

Coordinates of sector of circle

I know the coordinates of one point on a circle, this point is part of a sector. I know the angle of the sector at the centre of radius, I know the radius and I know the arc length. How do I calculate ...
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0answers
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12 points circle associated with a cyclic hexagon

When I research this problem A chain of six circles associated with a cyclic hexagon. I found the followings result. Let $ABCDEF$ be a cyclic hexagon. Let $A_1$ be any point on $AD$, the circle $(...
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1answer
26 views

Determining Locations of Circles to Optimally Cover a Polygon

I want to completely cover a region on a map(Continental US)/polygon with circles of a certain radius. Is there a way to determine the best locations and how many circles would be needed to completely ...
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0answers
29 views

How to calculate $\Delta$ in conic sections?

When learning conic section I learnt about $\Delta$. For any conic in general form : $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ Here $\Delta=abc +2fgh - af^2 - bg^2 -ch^2$ The conic is said to be ...
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1answer
29 views

How to determine if the implicit curve is closed?

Let the implicit equation $$F(x,y)=0, \quad (x,y)\in\mathbb{R}^2$$ defines a curve $\gamma$. The question is what properties must have the function $F$, s.t. the curve $\gamma$ be topologically ...
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2answers
42 views

Radius of inner circles given radius of outer circle and number of inner circles in circular fractal

I am trying to create a circular fractal in which each circle is composed by a given number $n$ of smaller circles. It would look something like this for $n = 8$: However, I don't know how to ...
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3answers
109 views

Circle containing other circle

Below is the question I am referring to: Two circles have the equations $x^2+y^2+\lambda x +c=0$ and $x^2+y^2+\mu x + c = 0$. Prove that one of the circles will be within the other if $\lambda\mu&...
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1answer
65 views

Area of a quadrilateral in which a circle can be inscribed using algebraic geometry

$\Delta POR $ has vertices $P(0,12),R(5,0)$ and $O(0,0)$. There exists a line $l$ cutting $PR$ and $OP$ at $A$ and $B$ respectively such that circles can be inscribed in $\Delta PAB$ and quadrilateral ...
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1answer
64 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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1answer
96 views

New Proof of Pythagorean Theorem (using inscribed circle)?

I was solving an easy problem for fun when I stumbled onto this, and was wondering if this was a correct and possibly a new proof of the Pythagorean Theorem. Given right triangle $\triangle ABC$, and ...
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1answer
30 views

Does it have to be a right angle?

Say you have a circle $O$ and a point on the circle $P$. From P, we create 2 points $A$ and $B$ on the circle such that $PA=X$, $PB=Y$, and the 2 points are on different sides of $\overline{PO}$ (...
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1answer
40 views

Prove that, given two chords on circle O labeled AB and CD, and given that arc AC and arc BD are equal, that AB=CD

Prove that, given two chords on circle $O$ labeled $AB$ and $CD$, and given that arcs $AC$ and arc $BD$ are equal, then $AB\parallel CD$. I understand that they are parallel, but I need help ...
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2answers
176 views

Determine the closest point along a circle's $(x_1, y_1)$ radius from any point $(x_2, y_2)$, inside or outside the radius of the circle.

I have a circle centered at point $(x_1, y_1)$ and another point at $(x_2, y_2)$. This point, $(x_2, y_2)$ may or may not be within the radius ($r$) of the circle. I wanted to create a line going from ...
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3answers
67 views

Area of a square whose one part is in circle

A square has two of its vertices on a circle and the other two on a tangent to the circle. If the diameter of the circle is $10$ cm, then what is the area of the square is? My solution: I figured ...
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2answers
33 views

Prove on Incenter and mid point.

Let the incircle (with center $I$) of $\triangle{ABC}$ touch the side $BC$ at $X$, and let $A'$ be the midpoint of this side. Then prove that line $A'I$ (extended) bisects $AX$.
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697 views

Relationship between two centers of circles in a Venn diagram

Let $S$ be a circle of 1 unit area. No part of circles $A$ and $B$ are outside the circle $S$. Let $n(S)=1$, $n(A)$, and $n(B)$ be the area of circle $S$, $A$, and $B$, respectively. For the given ...
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2answers
62 views

Solve using Butterfly Theorem.

Let $PT$ and $PB$ be two tangents to a circle, $AB$ the diameter through $B$, and $TH$ the perpendicular from $T$ to $AB$. Then prove that $AP$ bisects $TH$.
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1answer
30 views

Calculate point on circle perimeter with just radius, center point and X or Y offset?

How to calculate point on circle perimeter that is Y (or X) offset from another point on the perimeter? The center point, radius and offset are known. Sorry i have had no success googling this, maybe ...
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1answer
29 views

Finding position of 2D point constrained by “parent” point constrained to circle and rotation

I have the following 2D geometry question from a camera positioning problem: Point $P1$ (parent) can only be on a circle about the origin with given radius $R$. Point $P2$ (child)'s position is ...
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1answer
22 views

Calculating integral with area of half circle

I was recently told I can calculate this integral $ \int_{-R_0}^{R_0} \frac{\sqrt{{R_0}^2-r^2}}{2} dr $ using the formula for area of half circle can someone please show this to me I cannot do it
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1answer
18 views

Prove between Simson line & Nine point circle.

Prove that the Simson lines of diametrically opposite points on the circumcircle are perpendicular to each other and meet on the nine-point circle. I proved the first part of the problem but not able ...
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6answers
29k views

Parametric Equation of a Circle in 3D Space?

So, my dilemma here is... I have an axis. This axis is given to me in the format of the slope of the axis in the x,y and z axes. I need to come up with a parametric equation of a circle. This circle ...
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1answer
52 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. $(\cos a, \sin a)$...