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Questions tagged [circles]

For questions concerning circles. A circle is the locus of points in a plane that are at a fixed distance from a fixed point.

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How many circles can fit on the perimeter of $N$-gon

Given a regular $N$-gon with perimeter $P$ and circle with radius $R$. How many of these circles can fit on the perimeter of the $N$-gon without overlaping? $\frac P{2R}$ will give correct answer if ...
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2answers
38 views

Ratio between the width of the intersection of two identical intersecting circles and radius, when the intersection is $\frac{\pi r^2}{2}$

Or more visually, if all sections of the below diagram were equal in area and the circles are identical, what is the ratio of s and r, or what is s in terms of r. I came up with an equation using ...
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17 views

Equation of A Circle, given: tangent line, line containing the center and radius [on hold]

How do I find the equation of the circle if it is tangent to the line 3x-4y+12=0 and the center is on x+4y=-1 its radius is 5
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4answers
30 views

Finding lengths within circles when the circles are tangent to each other [on hold]

I have been attempting this question about circle geometry, but not entirely sure how to go about it. I was initially thinking about using trigonometry, but couldn’t get anywhere with it. Any ideas ...
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1answer
50 views

Real life applications of a circle? (Conics)

for my Math 2U assignment, we have to discuss real life applications of different conic sections. However, apart from the wheel, I cannot find or think of any other real life applications of the ...
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2answers
42 views

An identity associated with the centroid of a triangle

Four year ago, I am looking for a proof of my identity as follows: Let $ABC$ be a triangle, let $G$ be the centroid of $ABC$. Let $P$ be any point on the plane. Let $H, N, O$ on the plane such that: $...
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1answer
29 views

Calculating the break-even point between traveling around the inner / outer circle

Most days when I go to work, I ride a matatu (this is a Kenyan word. I am not in Kenya but the concept is the same). When I get off the matatu, I need to walk around the circle to get to the street ...
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2answers
56 views

Calculate the size of a polygon given the number and radius of circles that are on the perimeter

How can i calculate the size of a regular polygon if i have N circles of same size and all the circles must be on the perimeter of the polygon. Examples: 2 circles on hexagon 3 circles on ...
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3answers
24 views

Why's the base of the small arc length $r⋅dθ$?

As in this post, I'm trying to understand why area of a sector of a circle $= \dfrac{\theta r^2}{2} $ WITHOUT relying on $\dfrac{\theta }{2\pi} \pi {r}^{2}$ or integrals. To picture the emboldened ...
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1answer
13 views

Find value of x on a chord in the circle - which of the statements is true

in In the above image, is there a specific theorem that can be used to state which of the four statements must be true for x?
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3answers
33 views

Cartesian product of $S^1$ and $S^2$

Can anyone help me imagine $S^1 \times S^2$? I understand that $T^2=S^1 \times S^1$, but I don't know what to do with the spheres. I am not even sure if it's in 3D. Thank you!
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What is the center-point of a sphere called? [closed]

Based upon some quick searches I’ve discovered that the center of a sphere is literally called The Center of the Sphere. And the center of a circle is called the Circle’s Center. Have we really not ...
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1answer
31 views

Radius of a circle containing circles.

If, inside a big circle, exactly $n (n ≥ 3)$ small circles, each of radius $r$, can be drawn in such a way that each small circle touches the big circle and also touches both its adjacent circles (as ...
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1answer
32 views

Proving length of chords in a circle

$M$ is the midpoint of the chord $AB$ of circle $C(O,r)$. Show that if a different chord $ CD $ contains $M$, then $ AB < CD $ .(You may use Pythagoras's Theorem) Let $ \Delta CBM\sim \Delta ADM $...
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1answer
21 views

reflections of a point around the sides of a triangle: the area stays constant

Given a triangle $\Delta ABC$ and a point $P$, we define $P_A, P_B, P_C$ as the reflections of $P$ around $BC, AC, AB$ respectively. Now, $P_A, P_B, P_C$ are collinear if and only if $P\in(ABC)$. (1) ...
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3answers
51 views

why do we use $\pi \cdot d/2+d$ for semicircles

I'm in grade seven and right now we are learning perimeter of a semicircle, and I noticed we use the formula $\pi\cdot D/2+D$ (where $D$ is the diameter) instead of $(\pi+2)D/2$, why do we do the ...
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1answer
80 views

Maximum total area of n non-intersect circles?

Given n points on the x-axis, we give arbitrary radius for each point such that each constructed circle doesn't overlap another constructed circle from another point. Which means these circles do not ...
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2answers
34 views

Is it possible to cover all circle area with infinite lines starting from the center? [closed]

Is it possible to cover all area of a circle of radius r>0 with infinite lines starting from the center?
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3answers
238 views

Finding diameter of a circle using two chords and angle between them

Is it enough to find diameter of a circle using two arbitrary crossover chords with known length of each partition and angle between this two chords? If it's possible, how?
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3answers
48 views

Easier method of finding the equation of the circle circumscribing the triangle formed by 3 lines?

The equation of the circle circumscribing the triangle formed by the lines $y = 0, y = x$ and $2x + 3y = 10$ is? I know this can be done by solving two equations at a time and finding the vertex. ...
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0answers
23 views

A relation of the lengths of line segments in a circumscribed star pentagon.

A few days ago I posted a question and timon92 gave a beautiful solution. Using timon92's method, one can easily show that $ER\cdot BS\cdot CD=DE\cdot RS\cdot BC$ (as with the notations in the figure ...
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0answers
19 views

How to find the center of a circle moving and rotating in space having the coords of 3 points on the circle?

consider a circular plate with radius R , with three points on it at arbitrary coordinates on the plate area that form a triangle. Just imagine 3 holes on the plate area If I can have the new ...
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1answer
23 views

Chords in circle, finding angle [closed]

Circle image How can I find $\angle AOB$? (in terms of $\alpha$ and $\beta$)?
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3answers
37 views

find area of kite given side length

Circle with two tangent lines Above is the picture in question. A circle is given, center (-2,4) and a point outside the circle (0,10) is shown. Asked to calculate the area of the quadrilateral ABCD, ...
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1answer
21 views

Calculating the radius of this 'segmented' arc [closed]

I want to find the radius A of a circle such that each vertice of the 8 segments (including the 'ends' of this chain of line segments) touch the circle and the distance between these two 'ends' is 98....
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0answers
11 views

Maximum relative quotient of radii of circles surrounding other circle touching neighbors exactly once? [duplicate]

Assume we have one circle with radius $R$ centered at origo $(0,0)$. How big circles of radius $r$ can we put around it touching it and each neighbor in one and only one point each if we want to fit $...
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0answers
16 views

Three circles inscribed in a rectangle as shown in the diagram

3circles in a rectangle of width 6cm. The first circle of radius 3cm touches the three sides of the rectangle. The second of radius 2cm touch one side of the rectangle and touches the former and the ...
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1answer
26 views

A geometry problem about excircles and the tangency points

The excircle to side $BC$ of $\triangle ABC$ is tangent to lines $BC,AB$ and $AC$ at $D,E,F$ respectively. Let $P$ be the orthogonal projection of $D$ onto $EF$. Let the midpoint of $EF$ be $M$. Prove ...
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0answers
24 views

Is this “kissing incircles” property of a cevian through the Gergonne point (well-)known?

$\require{begingroup} \begingroup$ $\def\Ge{G_{\mathrm{e}}}$ A cevian through the Gergonne point $\Ge$ divides the triangle into two, whose corresponding incircles are "kissing" (mutually tangent). ...
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1answer
39 views

Please explain to me in simple terms what these circle intersections are all about.

So, say you got 4 circles intersecting this way: Now, I am looking for two things: A proof that each part of the circle which is in an intersection is 1/4 the size of the whole circle's ...
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Is this result already a known theorem in geometry?

I have been playing around with geometry and I found that: Let two perpendicular lines intersect at a point that is inside a circle. Then the area of the quadrilateral formed by the vertices made by ...
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3answers
32 views

Find the parametric equation of a circumference in a non traditional reference frame

I know that the parametric equation of a circumference center in (0,0) is: $y=\sin(\alpha)*r, x = \cos(\alpha)*r$. I try to write the equation of the same circumference with respect to the ...
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2answers
20 views

How to find the arc length between any two points (real numbers) on the circumference of a circle with center at the origin?

Suppose I'm given two points: (x1, y1) and (x2, y2) (which are real numbers) lying on the circumference of a circle with radius r and centred at the origin, how do I find the arc length between those ...
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1answer
40 views

The height of a section of overlapping circles.

Say I have two identical circles, both of radii of one, overlapping, as shown in the diagram below: In this diagram, x is the circumference of the circles, and the bit of the bottom circle which is ...
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2answers
30 views

Locus of the point of contact of tangent.

Let $A, B, C$ be three points on a straight line, $B$ lying between $A$ and $C$. Consider all circles passing through $B$ and $C$. The points of contact of the tangents from $A$ to these circles ...
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2answers
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Geometry problem only needed hint one more

I just want hint how to use the congruent condition..which involves an equation in X I guess And I obtained one equation $$y^2=x (16+x)$$
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3answers
53 views

$\triangle ABC$ with circumcenter $K$ has $AB=AC$ and $AD=BC$. Find $\angle BAC$.

In a triangle, $AB = AC$ and $D$ is its circumcenter. It is also known that $AD = BC$. Find the measure of the $\angle BAC$. Any tips for me being able to resolve the issue? I tried to let k ...
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0answers
49 views

Probability of two circles colliding in intersection area of two bigger circles

I have a small circle with area $A_s$ that is bound to be in a bigger circle with area $A_b$. The probability of the small circle being at a specific place in the bigger circle is: $$P = A_s/A_b$$ ...
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1answer
87 views

Given two normals on the circumference of a circle, find the change in height between the two.

I am provided with two unit normals which lie on the circumference of a circle. I am not provided with the points on the circumference that these normals stem from. I do know that the distance between ...
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0answers
28 views

Maximum number of nodes in a circle, with distance constraint

Given a circle $C$ with known radius $r$, I want to determine the maximum number of nodes in the circle, where there is a distance constraint between each two nodes equal to $s$, i.e. each two nodes ...
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2answers
35 views

Find the locus of the mid point of the chord

Find the locus of the mid point of the chord of the circle $x^2+y^2-2x-2y-2=0$ which makes an angle of $120^{\circ} $ at the centre. My attempt: Given equation of circle is $$x^2+y^2-2x-2y-2=0$$ ...
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1answer
281 views

Better method to solve a geometric problem.

This question is Q.13 of International Mathematical Olympiad Preliminary Selection Contest - Hong Kong 2019. $A$, $B$, $C$ are three points on a circle while $P$ and $Q$ are two points on $AB$. The ...
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5answers
38 views

Find all equations of a circle given two points

Find all the equations of circles $a (x^2 + y^2) + b x + c y + d = 0$ through two given points, $(-1, 2)$ and $(3, 1)$. I don't know how to approach this, I have to set up a matrix and solve but I'm ...
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3answers
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Pythagoras theorem is a^2 + b^2 = c^2 and a circle has an equation x^2 + y^2 = a^2 .Is there a relation between a right angle triangle and a circle?

I was just curious about the fact that whether such a relation exists when I came across the equation of a circle.(I maybe absolutely wrong) .
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2answers
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Stuck on step of the solution Circle theorems/Triangles

In $ABC$ right triangle, $AC=2+\sqrt3$ and $BC=3+2\sqrt3$. Circle goes on point $C$ and its center is on $AC$ cathetus, $C$ Cuts the circle in point $M$ and it touches $AB$ hypotenuse at point $D$. ...
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2answers
62 views

Showing that the unit circle has measure zero [duplicate]

How can I show that $\{x \in \mathbb{R}^{2}: |x| =1\}$ has measure zero using the definition of measure zero? I don't want to use the interpretation of measure in $\mathbb{R}^2$ as area (and then ...
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1answer
44 views

Difficult circle geometry question, proof sum of 2 sides ratio equals 1

I have been working on this question for hours on end but not even come close to solving. I have found 2 pairs of similar triangles and an isosceles triangle, and tried to equate the ratio of sides, ...
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1answer
20 views

Using the discriminant to find the equation of tangents to a circle

I was working on a geometry question and took a really long winded route to get an answer. A worked solution I found for it used the discriminant but I don't understand how. The question was this: ...
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4answers
303 views

Does the unit of measure matter when you are solving for the diameter of a circumference?

I understand that the circumference of an object should be divided by $\pi$ when searching for the diameter. What I don't understand is what unit of measurement should I use, inches, cm, mm? If I ...
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1answer
38 views

Two circles of equal radius, one “passes over” the other: the intersection area grows and shrinks non-smoothly?

Two circles of equal radius "r" (assume r=1 for simplicity); one "passes over" the other. They start by touching, having an empty intersection, and then slowly the intersection grows, until the ...