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.

4,045 questions
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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$\triangle ABC$ 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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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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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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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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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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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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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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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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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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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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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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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: ...
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 ...