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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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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
116 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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37 views

Position of circle against two other circles to satisfy a condition

consider three different circles $C_1(\{x_1,y_2\},r_1)$, $C_2(\{x_2,y_2\},r_2)$, and $C_3(\{x_3,x_3\},r_3)$ and $P_1, P_2, P_3$ are their corresponding centers. How to find best $P_3(x_3,y_3)$ such ...
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1answer
37 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 ...
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Prove that $AR$ passes through midpoint of $BC$

Consider $\Delta ABC$ is acute triangle, $O$ is circumcircle of $\Delta ABC$. $AD$ is angle bisector of $\angle BAC(\text {D} \in \text {BC})$, $E;F$ are respective in $CA,AB$ such that $CE=CD$ and $...
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Has the braiding pattern of circles with points infinitesimally close to the center been studied?

Suppose I have $n$ circles in $\mathbb{R}^2$ centered at $(\epsilon^i_x,\epsilon^i_y)$ infinitesimally close to the center, with radii $r_i=1+\epsilon^i_r$ infinitesimally close to 1. In general this ...
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1answer
23 views

Number of isosceles triangles formed with 12 equally spaced points lying on the circumference of a circle [on hold]

How many isosceles triangles can be formed with 12 equally spaced points lying on the circumference of a circle? The answer should be 52, but I have no idea how to solve it. Please help!
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2answers
130 views

How can i Prove that the gray area is the same as white area? [duplicate]

A circle is cut into 8 parts, each part has the angle 45 degrees from an arbitrary point. how to prove that the white area is the same as the Gray area? I just want any hint for solving this question....
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3answers
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Finding *two* equations of circles given 2 common points and a common tangent.

Suppose we're given 2 points on a circle, $A(1,3)$ and $B(2,4)$, and its tangent being the $y$-axis. We're asked to find the equation of the circle. If I use the standard procedure $$(x_n-h)^2+(y_n-k)...
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1answer
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How do you find the eccentricity of an ellipse given the intersection points with a circle that shares the center with the ellipse?

I was wondering how to find the equation of an ellipse with points that intersect a circle with the same center. I attempted to solve it using these following parameters: (a,b) , (c,d) , (e,f) , (g,h) ...
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2answers
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distance between the radii of two intersecting congruent circles with virtually no info

intersecting circles There are two intersecting congruent circles, and I only know the length of the small arcs (coloured in red the image above), is there any way to find out the distance between ...
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1answer
32 views

Help with proving the following points are collinear

Let BC be the shortest side of $\triangle$ABC. Let P be a point in AB such that $\angle$PCB=$\angle$BAC and Q be a point in AC such that $\angle$QBC=$\angle$BAC. Prove that the line that passes ...
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0answers
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Rational points on circle with hcf of Num and deno 1

Let $(\frac{{{p}_{1}}}{{{q}_{1}}},\frac{{{p}_{2}}}{{{q}_{2}}})\,\,and\,\,(\frac{{{a}_{1}}}{{{b}_{1}}},\frac{{{a}_{2}}}{{{b}_{2}}})$ be any two rational points on the circle ${{x}^{2}}+{{y}^{2}}=1$ ...
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1answer
45 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
46 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
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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
26 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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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
25 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 [closed]

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
41 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
30 views

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
44 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
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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
41 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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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: Center - (x,y) Radius - r Heading angle - s - the ...
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2answers
36 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
149 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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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
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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
35 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
22 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
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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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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. ...
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3answers
196 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
28 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
53 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 ...
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1answer
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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 ...