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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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1answer
36 views

a circle and a parabola have 3 intersection points

Is it possible that a circle and a parabola on a euclidean plane have 3 intersection points and the center of the circle does not lie on the axis of parabola?
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1answer
42 views

Show that a triangle is equilateral

A circle crosses the sides of a triangle, dividing each of them into three equal parts. Prove that the triangle is equilateral. I think that the best way is to show that $\angle BAC = \angle ABC$, ...
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3answers
29 views

Find the intersecting point of yaxis for the xaxis of a circle

I need to find the intersecting points of x and y axis in a circle for the given radius of 40. So I found the formula $(x-h)^2 + (y-k)^2 = r^2$, center of the circle is $(0,0)$. X-axis is from $-5$ to ...
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2answers
25 views

Derivation of Family of Circles Meeting at Two Points [duplicate]

We know the formula of "family of circles intersecting at two points" as follows: $$x^2 + y^2 + D_1 x + E_1 y + F_1 + \lambda (x^2 + y^2 + D_2 x + E_2 y + F_2) = 0, \qquad \lambda \setminus \{-1\} \in ...
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0answers
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plane edge intersections embedded in higher dimensional space

Let's say we have some D-dimensional euclidean space, and we have some circles of dimension 0 to D-1 (circle dimensionality meaning the minimum number of vectors needed to fully define it, so a ...
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0answers
23 views

Compute the radius and the central coordinate (x, y) of a circle constructed by three given points on the plane surface

I need you to explain the mathematics behind the code bellow. What is s, what are those formulas for px and py and generally, what logic are we following to find the answer here? ...
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1answer
22 views

Mobius transforms - inversion sends line/circle -> line/circle

I'm trying to prove that the inversion mapping $f(z) = \frac{1}{z}$ sends circles or lines to circles or lines. Apparently the set $$\{z \in \mathbb{C}: |z-a|^2 = r^2 \}$$ describes either a circle ...
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20 views

Ways in which circles cut each other [on hold]

What will be the number of ways n circles cut each other, for n like 2,3 it is doable but is there any pattern which can be used to build a program or some sort of a series to follow up ?
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2answers
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GRE Geometry — Overlapping Circles with an enclosed figure.

Hi I'm having trouble solving this problem from a quantitative reasoning question on the GRE. The diagram and the word problem are shown above. I think I'm lost as to the properties a parallelogram or ...
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4answers
39 views

Trig Equation of a Circle

I was trying out derivatives of inverse functions(I'm new to that) which led me to this. I graphed the equation $y=\sin\left(\cos^{-1}(x)\right)$. This is a half circle and squaring both sides gives a ...
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2answers
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A circle inscribed in a trapezoid; $\angle BCH$

$ABCD: AB ||CD, AB>CD, AD=BC$ $k(O)$ inscribed $DH \bot AB,H \in AB$ and $\angle ADC = \gamma$ $\angle BHC, \angle BCH =$ ? I have tried to show that $\triangle BCH$ is isosceles, but when I ...
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3answers
40 views

An Extra Solution

I hope you're well, I was finding the points of intersection of $$ x^2 + (y-1)^2 = 1 \quad \text{and} \quad y = 1-x^2 $$ If I rearrange the formula of the circle to $$ (y-1)^2 = 1-x^2 $$ then ...
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0answers
39 views

A circle inscribed in a quadrilateral

A circle is inscribed inside a trapezoid $ABCD$ ($AB || CD$). $M, N, P, Q$ are the midpoints of the sides $AB, BC, CD, AD$, respectively. If $AD = d$ and $BC=c$, express the perimeter of $MNPQ$. (...
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1answer
37 views

Find the diameter of a circle subtended by an angle

The question doesn't state whether its subtended at the center or circumference, but I not sure if it matters The sector a circle subtended by an angle of $22.5$ degrees has an area of $\frac{9\pi}{4}...
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1answer
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Maximum Regions Vees Can Divide a Circle

The Circle Division by Lines problem (link) asks into how many regions, at most, one can divide a circle (or: the plane) with $n$ chords (or: lines). I am wondering about a similar question, but for ...
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0answers
41 views

Half circles collision

How to find if 2 half circles collide ? Hi, I have 2 half circles, circle1 and circle2, and I want to know if they collide. Half circle details: Have a center - (x,y) Have a radius - r Have a "...
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0answers
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how to find points on two circles such that these points are separated by a specific known distance? [closed]

I have two circles whose centre points and radii are known. I want to find the set of point pairs(P1,P2) such that P1 lies on circle1, P2 on circle2 and the distance between P1 and P2 is a known ...
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2answers
35 views

Given a circle $A$ of area 1 centered at $\{0,0\}$, give conditions that another circle $B$ of known area <1, lies totally within $A$

Given a circle $A$ of area 1 centered at $\{0,0\}$--so, of radius $\frac{1}{\sqrt{\pi}}$--give conditions on the possible location of the center $\{x,y\}$ of another circle $B$ of known area $\pi r^2 &...
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2answers
63 views

Finding area common to two circles

Having trouble finding the area common to the two mats. The correct answer is $99.1cm^2$
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1answer
45 views

Perimeter of two overlapping circles

How do I find the perimeter of these overlapping circles? The correct answer is 99.5
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4answers
45 views

Why is my approach for showing $r^2 \frac{\theta}{2}$ equals the area of a circular sector incorrect? Do we need calculus?

I know that the area of the sector of circle can be computed using the following two formulas $$\pi r^2 \frac{\theta }{360} \space \space \text{ (degrees case)}$$ $$or$$ $$r^2 \frac{\theta }{2} \...
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3answers
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Is the circle homeomorphic to a $6$ petal rose?

I am trying to explain that a circle is homeomorphic to a $6$-petal rose with the standard topology of $\mathbb R^2$, for that i'll need to explain (just in words, it is not necessary to come up with ...
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3answers
148 views

Integral of $x^2$ over $x^2+y^2 ≤ a^2$

This is in my lecture notes, I understand that $x=rcos\theta$ so therefore its the integral of $x^2=(rcos\theta)^2$ but why is there an r in $rdrd\theta$ in there? Any help would be appreciated.
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circular measure regarding internal and external contact [closed]

Two circles of equal size, each of radius 5 cm, touch each other externally. Both of them touch another circle, centre O and radius 20 cm, internally as shown in the diagram. calculate the area of the ...
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0answers
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Reflect point inside Poincarè Disk Model

I have a question: is there a formula which map points onto the Poincarè Disk Model starting from points which are outside the unit Disk? For example, I have a point p with norm = 2, so it is outside ...
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0answers
48 views

If $R:S^{1}\rightarrow S^{1}$ is a irrational rotation, $\{R^{n}([x])\}$ is dense in $S^{1}$ for all points. [duplicate]

Let $\alpha$ a irrational number, and $R:S^{1}\rightarrow S^{1}$ the irrational rotation, i.e., $[x]\rightarrow[x+\alpha]$. I need to prove that, for all $[x]\in S^{1}$, the set $\{R^{n}([x])\}$ is ...
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1answer
34 views

Two circles with two common intersect points

I have two circles which intersect and have two common intersections. The larger circle has radius 5 and the smaller has radius 3. The distance between the centre of the two centres of the circle is ...
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1answer
38 views

Prove that three common chords are concurrent

Three circles intersect each other as shown. Prove that the three common chords are concurrent. Now the book does this by proving that the chord out of E and through M is the same for circle (2) and ...
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1answer
21 views

Inscribed trapezoid in a circle

Find the angles of an inscribed trapezoid (in a circle) $ABCD$ ($AB||CD$) if $\angle ABD = 63^\circ$. Any trapezium in a circle is an isosceles trapezium, so $AD = BC$, thus $\newcommand{arc}[1]{\...
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4answers
728 views

What is the radius of the circle in this problem? [closed]

Context: This is one of the entrance exams for one university. The exercise gave the following information: $AB$ and $CD$ lines are parallel to each other. The length of $CD$ is 4 and the length ...
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0answers
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Find sub areas of a function in a circle

I have a cellular signal calculation function, which calculates the signal given the distance from the antenna. Without the constants, the function is basically: $f(d)=1/(d^α)$ where α is a parameter. ...
13
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3answers
191 views

Circles generated by three-fold iterations $f(x)=\frac{1}{1-x}$

I came across a weird property of the function $f(x)=\dfrac{1}{(1-x)}$ Observe the following: $$f(x) = \frac{1}{(1-x)}, \quad\quad f^2(x) = f(f(x)) = \frac{(x-1)}{x}, \quad\quad f^3(x) = f(f(f(x))) ...
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1answer
27 views

Angle created by three distincts random vertices

Assume you have a regular polygon ( $n$-sides). and Let $A=\{ x_0, x_2, \cdots , x_{n-1} \}$ be vertices of the polygon. My Question is: Are there is any formula that tell us what is the angle ...
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2answers
49 views

Geometry problem (Inscribed angle theorem, circumscribed circle)

Let A and B be two different points. Show that the points P are such that the angle APB is 90 degrees and creates a circle. Decide the the radius and mid point of the circle. I have problems proving ...
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1answer
51 views

Finding the center, radius of a circle when there is a constant in front of the variables [closed]

$$4x^2+(y-2)^2 = 4$$ How to find the center and radius of a circle when there is a constant in front of the variables?
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Radius of a circle in terms height of right triangle

Here are the constants and the arc length s=R*theta is considered constant Given only H and the arc length between theta How can you find the radius of a circle? I was able to find a relationship ...
3
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1answer
37 views

Triangle centers Orthocenter

An acute $\triangle ABC$, inscribed in a circle $k$ with radii $R$, is given. Point $H$ is the orthocenter of $\triangle ABC$ and $AH = R$. Find $\angle BAC$. (Answer: $60^\circ$) $AD$ $-$ ...
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1answer
47 views

Find triangle with given orthocenter and an apex on a circle

We are given $k(O; r)$, $A \in k$ and $H : OH>r$. Find points $B$ and $C$ ($B, C\in k$) such that $H$ will be the orthocenter of $\triangle ABC$. I am trying to see what is the point $F$ ($AH \cap ...
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1answer
52 views

Inversion in circle

Let $C$ be a circle with the middle point $O$ and the radius $r$, we say that the points $P$ and $P'$ are inverse points with respect to $C$ if: $1.$ $|OP|·|OP'|=r^2$ $2.$ $P$ and $P'$ are on the ...
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1answer
64 views

Question about loops(closed paths) in $\mathbb{S}^{1}$

Let $$\alpha \left(s \right) =\left( \cos{2\pi s},\sin{2\pi s}\right)$$ and $$\beta \left(s \right) =\ \left( \alpha \land\left( \alpha \land \overline{\alpha} \right)\right)\left( s \right)$$ with $...
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3answers
43 views

How does the Pythagorean theorem describe a circle?

The Pythagorean theorem states, for a right triangle with legs $a,b$ and hypotenuse $c$, $$a^2+b^2=c^2$$ By replacing $c$ with $r$, radius this equation becomes the equation of circle at centre $(0,...
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1answer
28 views

How do i calculate the angle $\theta_L$?

I need to find the angle $\theta_L$ in the attached sketch. The variables I know are: $R_L$, $R_G$, and $\theta_G$. So i need a formula for $\theta_L$ in terms of these. $R_g$ is also the distance ...
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0answers
34 views

Checking circle theorems question

I was just wondering if these answers were right. I am pretty sure that part a is correct, but part b I am not as confident with.
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1answer
30 views

What is the length of the arc on the unit circle subtended by an angle of 120 degrees? Show all work.

What is the length of the arc on the unit circle subtended by an angle of 120 degrees? Show all work. 2/3 1/3(pi) 2/3(pi) pi I used an equation where the central angle equals the arc length divided ...
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1answer
15 views

Is the intersection of multiple circles that all intersect a specific point always convex?

Assuming I have some circles whose mutual intersection area is > 0, and each of the circles' edges intersect the same point, e.g. as shown in this picture: Will the mutual intersection set (edge + ...
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3answers
47 views

Formula for average width of a circle [closed]

I'm trying to make a costing calculator for engraving a circle on acrylic using a laser. The laser can move at 150mm/sec, and lightly cuts lots of parallel lines spaced 0.05mm apart. If I engrave a ...
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0answers
21 views

Maximum number of common chords that are existent between two Conics

The maximum no. Of common chords between a circle and a parabola is 6. this is because they can have at most Four Points of intersection. However I have doubt regarding other combination of conics. ...
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1answer
52 views

A geometry problem : A circle, centred at $I$ has diameter less than the length of the segment. [closed]

Let $AB$ be a line segment with midpoint $I$. A circle, centered at $I$ has diameter less than the length of the segment. A triangle $ABC$ is tangent to the circle on sides $AC$ and $BC$. On $AC$ a ...
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5answers
319 views

how to find the equation of a circle given points of the circle [closed]

Find the equation of the circle which passes through points $P(-6,5)$, $Q(2,1)$ and has its centre lying on the line $y=x+4$ I would appreciate any working given, as I do not understand how to find ...
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1answer
86 views

Circle and triangle

A circle $k(O)$ with diameter $AB$ is given. Lines $PC$ and $PD$ touch $k$ ($C, D \in k)$. $AC$ $\cap$ $BD =K$. Show that $PK \bot AB$. This is what I've done for two days. I will be very grateful if ...