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.

learn more… | top users | synonyms

1
vote
1answer
21 views

Probability of being in a circle, given normal

Let's assume a bivariate normal distribution with center $\mu$ and covariance matrix $\Sigma$. Let a circle $C$ be given as $C=\{x\in\mathbb{R}^2:||x-\mu||\leq R\}$. I would like to calculate the ...
1
vote
2answers
38 views

Find area of this circle

Given the circle O with perpendicular diameters and a chord, find the area of the circle if $EF = 8"$ and $DE = 20"$ inches.
0
votes
0answers
13 views

More generalization of the Sawayama lemma

Let $ABC$ be a triangle, $P$, $Q$ be two isogonal conjugate. $AP$, $AQ$ meets (ABC) at $D, E$ respectively. Two lines through $D, E$ meet (ABC) at $T, N$ and meet BC at $G, H$ respectively. Let $PG, ...
0
votes
2answers
26 views

Find the distance between the center of an unit circle and an internally tangent circle, which's tangents meet at the outer's center in a 72° angle.

There is a circle c, that has a center O, and a circle d that's internally tangent to it. If there are two tangents of d that meet at O, making a $72°$ angle, what's the shortest distance between the ...
0
votes
3answers
25 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 $?$...
0
votes
2answers
71 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 ...
2
votes
0answers
44 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.
1
vote
1answer
33 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) ...
0
votes
2answers
24 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? ...
0
votes
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?
0
votes
2answers
37 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. ...
3
votes
2answers
64 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 ...
0
votes
0answers
28 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 ...
3
votes
1answer
43 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 ...
1
vote
1answer
21 views

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,...
1
vote
1answer
28 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 ...
-1
votes
2answers
129 views

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

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 ...
1
vote
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 $AB$...
0
votes
1answer
24 views

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

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 ...
1
vote
1answer
39 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. ...
0
votes
1answer
23 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 ...
2
votes
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 $r=r_0$ ...
0
votes
1answer
50 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=...
1
vote
0answers
33 views

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 $(...
4
votes
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 ...
3
votes
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 ...
1
vote
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 ...
2
votes
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 ...
1
vote
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 ...
2
votes
3answers
110 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&...
0
votes
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 ...
5
votes
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 ...
0
votes
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 $\overline{PO}$ (...
3
votes
3answers
68 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 ...
0
votes
1answer
41 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 ...
2
votes
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 $AX$.
1
vote
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 ...
0
votes
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 ...
1
vote
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
2
votes
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 $TH$.
0
votes
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 ...
1
vote
1answer
67 views

A geometry problem hinting similarity of triangles .

I recently came across a geometry problem , published in an local magazine(publishing at high school and under graduate level) and was under Difficulty : Hard sub heading. Consider a $\triangle ABC$ ...
0
votes
0answers
11 views

Is a circular restriction possible (will elaborate)

When making rational functions, you may get various kinds of restrictions (as the difference in power increase the difference is the shape of restriction) so if by stating an implicit function that ...
0
votes
3answers
36 views

Prove for Pedal & Isosceles triangle.

The tangents at two points $B$ and $C$ on a circle meet at $A$. Let $A_1B_1C_1$ be the pedal triangle of the isosceles triangle $ABC$ for an arbitrary point $P$ on the circle, as shown below. Then ...
0
votes
1answer
61 views

Is ellipse intersecting with circle?

I have circle given by center coordinates and radius, and ellipse with center coordinates, $r_x$ and $r_y$. I want to check if the ellipse is inside the circle( meaning their bounds can collide). How ...
0
votes
1answer
28 views

Counting number of points making angle < 90

I have a around 1000 points and 1000 segments in the form of $(x_1, y_1, x_2, y_2)$ meaning the segment starts at coordinate $(x_1, y_1)$ and finishes at $(x_2, y_2)$. For each line i want to know how ...
0
votes
2answers
44 views

Plotting an arc with no center point - a practical solution please!

I need a mathematical solution to a very practical problem (laying a patio). The attached will hopefully explain. The center of the circle for the arc we wish to have is inaccessible (ie in the ...
3
votes
4answers
202 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 $2r$?
2
votes
2answers
100 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 ...
0
votes
3answers
39 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)/ ...