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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18 views

The locus of points of the form $ae^{i\theta}+be^{i\phi}$

Let $a$ and $b\in\mathbb{R}^{>0}$ be two positive real numbers. What is the locus of points of the form $ae^{i\theta}+be^{i\phi}$ where $\theta$ and $\phi\in[-\pi,\pi)$? Does it have any specific ...
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Prove that A₁D, B₁E, C₁F intersect at the same point

Let ABC be a triangle.AA₁, BB₁, CC₁ are the angle bisectors of the triangle. ω is circumcircle of ABC. ω∩AA₁=A₂, ω∩BB₁=B₂, ω∩CC₁=C₂. Circumcircles of AB₁B₂, BC₁C₂, CA₁A₂ intersect with AB,BC and CA at ...
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Tantalising Geometry!

A triangle with side lengths $5$, $6$ and $9$ has a circle inscribed IN IT! (ignore the wording below) What is the radius of the circle? I'm not sure where to start and, as well as hints/an answer, ...
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Minimize area of n-gon circumscribed around unit circle

Given regular unit circle and a n-gon, circumscribed around this unit circle. I need to minimize area of n-gon. Also i need to find the limit of n-gon area with $n\rightarrow \infty$. Intuitively i ...
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4answers
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What is the size of each side of the square?

The diagram shows 12 small circles of radius 1 and a large circle, inside a square. Each side of the square is a tangent to the large circle and four of the small circles. Each small circle touches ...
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3answers
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Can arc be measured by degree? [closed]

I saw a math problem that says the measure of an arc of a circle is $40^\circ$. My question is how can an arc be measured by degree? What is it called? Please make me understand.
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1answer
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Prove that A(FKP)=A(EDP).

ABCD is circumcircled in W. It is trapezoid. Mid points of parallel sides, BC and AD are E and F respectively. AB and DC intersect at P, W(circle) and BF intersect at K. Then prove A(FKP)=A(EDP) Where ...
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Where the shaded region is added to the perimeter

I have some problems finding the perimeter of the unshaded region as the picture below. $PQ = QR = PS$ is $5$ cm. $SR$ is $11$ cm. The radius of the circle is $1.2$ cm. Perimeter of the trapezium: $5\...
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Directed angles problem: For $A$, $B$, $C$ on $\bigcirc O$, show that $\angle OAC=90^\circ-\angle CBA$

Directed angles problem from Evan Chen's book: Points $A$, $B$, $C$ lie on a circle with center $O$. Show that $$\measuredangle OAC=90^\circ-\measuredangle CBA$$ (This is not completely trivial.) ...
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How can we speed up and improve circle-approximation using symmetries?

I am currently on the hunt for a way to approximate a half-circle function $$t \to f(t) = \sqrt{1-t^2}$$ By means of truncated power series (polynomials). I found several issues with a previous ...
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Proving the degree of the antipodal circle function.

I was asked to determine the degree of the antipodal circle function $f(\theta)=\theta + \pi$. I believe the degree of the function is $1$, through pictures and illustration, but I will like to prove ...
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Question about homeomorphism

can someone give a help on this? Let $T:S^1 \rightarrow S^1$ be given by $T(\theta) = \theta + \omega + \epsilon \sin (\theta)$ where $\omega$ and $\epsilon$ are constants. Prove that $T$ is a ...
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Find the radius of the circle given in the picture below.

This is the image of the question. I am not able to get how to find the radius. Please help with that. This is my try. I can't proceed now after it. Thanks
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Straightedge-and-compass construction of the “kissing circles” for three given circles

Let $C_1,C_2,C_3$ be three mutually tangent circles. Call the circle tangent to all of them (that is, intersecting each at one point) and enclosed within the region between them their kissing circle. ...
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Finding the area of the sector in the disc cut out by two rays.

Working on the book: Lange, Serge. "Basic Mathematics" (p. 119, exercise 1). Finding the area of the sector determined by the angle in exercise a (area of disc is $60in^2)$: Let $S$ be the sector ...
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Right tangential trapezoid

In a right tangential trapezoid $ABCD$ $(AB\parallel CD)$ and $AD\perp AB$ the incircle is $k(O).$ Find the area of the trapezoid if $CO=6$ and $BO=8.$ The triangle $BOC$ is a right triangle and by ...
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1answer
28 views

Circle homeomorphism

Simple problem, searchig for a solution. Let $A$ be a proper, closed and path connected subset of the $S^1$, then $A \cong [0,1]$. So currently I am trying too use the fact that $f:[0,1] \rightarrow ...
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Tangent to the circle concentric with the circle, $2x^2+2y^2-6x-10y=183$

Let $C$ be the circle concentric with the circle, $2x^2+2y^2-6x-10y=183$ and having area $(1/10)th$ of the area of this circle. Then a tangent to $C$, parallel to the line, $3x+y=0$ makes an intercept ...
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Prove that $AD\cdot BD \cdot CD \leq \dfrac{32}{27}$ where $ABC$ is a triangle of circumradius 1 and $D\in (BC)$.

Let triangle $ABC$ of circumradius $1$ and $D$ a point on side $(BC)$. Prove that $$AD\cdot BD\cdot CD\leq \dfrac{32}{27}.$$ My idea. By letting $\alpha = \dfrac{BD}{BC}$ (of course $0<\alpha &...
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1answer
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Is there a way to represent functions using circles, similar to how Taylor series work?

I was curious whether or not there is a method of representing a function as an infinite amount of circles multiplied or added or something? Similar to a Taylor series, such that the method would ...
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What is the relationship between the area and circumference? [duplicate]

I know the area of the circle is πr² and the circumference is 2πr, but what is the relationship between these things? The other page doesn't really make me understand it. ...
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Find the equation of all circles tangential to the lines $y = 0, x = 0$ and $y = - x + 2$

I have a question, to find the equation of all circles tangential to the lines $y=0,\,x=0$ and $y=-x+2$. There should be $4$ circles. I understand so far that circles take the form $$(x - h)^2+(y - k)...
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ABC is an acute-angled triangle whose altitude from the vertices meet the circumcircle

Let ABC be an acute-angled triangle. Let the altitudes from the vertices A, B, C meet the circumcircle at P, Q, R whose corresponding complex numbers are $z_1,z_2$ and $z_3$ respectively. If is $\...
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Check if segment crosses circle

I have 2 points $p_1$ and $p_2$ in a $2D$ plane. There are $N$ balls of radius $r$ scattered in the plane. I want to know which balls will be crossed by the segment $p_1 \rightarrow p_2$. I did it ...
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1answer
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How to find the locus traced by the Centre of a circle

If circle $x^2$ +$ y^2$ +2gx +2fy -12 = 0 is orthogonal to the circle $x^2$ + $y^2$ -4x -6y -2 = 0, the find the locus of the centre of the first circle I have tried using the relation for ...
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Find the area of the entire shape.

ABCD is a straight line. ABE is a sector of a circle with center B. CED is a sector of a circle with center C. Angle ABE is a right angle. The length of AB is r and angle ACE is π/4 radians. If r=10cm,...
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Smallest value of a graphed circle?

So I have a question saying, "Find the smallest value of $$ x^2 + y^2 -2x + 6y + 3= 0." $$ I know how to make this into a circle equation $$(x-1)^2 + (y+3)^2 =7$$ (if I'm not wrong) and graph it, ...
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Calculate the area of a sector of a circle? [My answer is different to the textbook answer, who is correct?]

Ok, so the link attached shows my answer with working out, the textbook's answer and the question. If someone could explain if my answer is correct or if the textbook's answer is correct? https://...
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can a complex math equation can create multiple closed area?

example : equation of circle can easily represent case A. can some single complex mathematical equation can create case B ?
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Detect correct ball that will be hit first

Suppose there are some static balls of radius $r$ in a $2D$ table. I will put one of them (let's call it $b$) with velocity $v$. My question is: how to detect which ball will be hit first? I couldn't ...
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1answer
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Orthogonal circles in Modern Geometry

Si una recta trazada por un punto de intersección de dos circunferencias corta nuevamente a las circunferencias en $P$ y $Q$, respectivamente, desmuéstrese que las circunferencias con centros P y Q, ...
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3answers
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The base of an isosceles triangle; given leg and radius of circumcircle

An isosceles triangle $\triangle ABC$ is given with leg $AC=5$ and $R=\dfrac{25}{6}$ of the circumcircle. Find the base of the triangle. This was my first sketch. The triangles $AHC$ and $OMC$ are ...
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3answers
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Finding the radii that maximizes and minimizes the area of four inscribed circles in an equilateral triangle.

An equilateral triangle with side length $1$ unit contains three identical circles $C_1$, $C_2$ and $C_3$ of radius $r_1$, each touching two sides of the triangle. A fourth circle $C4$ of radius $r_2$ ...
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Why does the polar graph of 1/x form a straight line at the beginning?

When $$r=\frac{1}{\theta}$$ is graphed in polar form, there is a straight line at the beginning. Why does this happen? What is the relationship between $$\frac{1}{x}$$ and circles?
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1answer
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Find the equation of a circle given two points and its radius - based on the coordinates of another circle

There is a circle of radius 187.5m and centre $(25.23,188.6)$. It takes the form $(x-25.23)^2 +(y-188.6)^2=187.5^2$ There is another circle whose centre is unknown and has a radius of 1750m. This ...
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The base of an isosceles triangle; given the radius of the inscribed circle and perimeter

On the sketch below $AC=CB$ and $OD=\dfrac{4}{10}CD$. If the perimeter of $\triangle ABC$ is $P_{\triangle ABC}=40$, find the length of the base $AB$. Let $AC=BC=a$ and $AB=c$. Since the triangle is ...
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Is 5.6 radians equivalent to 1.2 theta?

So I was studying some circle properties in Khan Academy and this problem basically states that if: $S=r\theta$ then find a formula for $S$ in terms of $r$. Now in the problem it says that the angle ...
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2answers
36 views

If $C_1$ has a radius of $10cm$ then show that the sum of the areas of all these circles is $\frac{25π}{3\sqrt2-4}cm^2$.

QUESTION: Let {$C_n$} be an infinite sequence of circles lying in the positive quadrant of the $x$-$y$ plane with strictly decreasing radii and satisfying the following conditions. Each $C_n$ touches ...
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1answer
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Doubt for the area of shaded region for trapezium

I wonder if I can use the area of trapezium for the shape CAOB? I've abit of doubt for the arc slope as often, the slope of the trapezium appears to be a straight line. For part a) I'm just checking ...
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How to find the arc centre and radius given the arc start point and arc end point and arc direction?

I know the arc start point and the end point. It is separated by a height/distance as shown in the figure.The blue line end points are the arc start and end points respectively. How can i drawn a arc ...
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Work done by a function along a circle

Find the work done in moving a particle around a circle C in the XY plane if the circle has the centre at origin and radius 2 and force F is expressed as F=(2X-Y+2Z)i+(X+Y-Z)j+(3X-2Y-5Z)k In the ...
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1answer
42 views

The legs of a right triangle with given hypotenuse and radius of inscribed circle

Triangle $\triangle ABC$ is a right triangle with $\measuredangle ACB=90^\circ$. Let $AB=c$ and the radius of the inscribed circle be $r$. Find the catheti and the area of the triangle $\triangle ABC$....
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Calculating diameter of neighboring circles

I have one circle C1 with known diameter d1. That circle is surrounded by N circles (C2) of unknown diameter d2 where each of C2 circles touches two neighboring C2 circles and C1 circle in one point. ...
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2answers
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The catheti of a right triangle with given hypotenuse and radius of inscribed circle [duplicate]

Triangle $\triangle ABC$ is a right triangle with $\measuredangle ACB=90^\circ$. Let $AB=c$ and the radius of the inscribed circle be $r$. Find the catheti and the area of the triangle $\triangle ABC$....
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Need some Worked Solutions to Circle Geometry questions that use 3 to 5 different theorems. [closed]

I am helping with some practice questions for a friend but I am terrible at circle geometry. Some help would be appreciated Edit: I am looking for some questions that require the use of 3 to 5 ...
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1answer
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Prove or disprove that the points are aligned (Witch of Agnesi)

It's a little problem that I cannot disprove or prove . We work with the following picture : First we draw a circle and his tangent at point $D$ and $E$ .Secondly we draw the radius or the line $DE$....
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41 views

Circle with rectangles inside

In this figure our aim is to find radius of the circle. Note that only $4$ vertices of rectangles are on the circle. My try: Since the top rectangle side and bottom rectangle side are parallel, From ...
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1answer
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If a circle cuts an ellipse (both centered at the origin) at four distinct points then find the maximum value of the acute angle formed.

QUESTION: Let $E$ be an ellipse with centre at origin $O$ and the major and minor axis to be $2a$ and $2b$ respectively. Let $\theta$ be the acute angle at which $E$ is cut by a circle with centre ...

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