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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What's the average distance between two discs in the plane?

Consider two discs in the plane of radius $r$ and $s$, with centers separated by a distance $l$. If we choose a point uniformly at random from each disc, what is the expected distance between the two ...
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Construction involving regular polygons inside a circle

Let's make a construction involving regular polygons: ► First, we begin with a equilateral triangle, with side $\ell_3 = 1;$ ► After, we draw a square on the middle point each side of the initial ...
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Iterating three tangent circles using Malfatti Circles

First, construct three tangent circles (blue circles), then construct the triangle joining their centers. Then construct three Malfatti Circles for this triangle (green circles). Go on. What I'm ...
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Stacking circles

When I tried to stack 21 circles of radii $(30, 31, 32... 50)$ on top of each other in a tube (ID of $100$ wide), I thought they would reach the same height regardless of the order, however I was ...
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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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The reverse pizza problem .

The pizza problem is a fairly well-known problem which sounds like this : You have a circular pizza and you need to cut it such that you and your friend would both receive half of the pizza . ...
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Pinwheel- perimeter of semicircular region

Above, we have a larger circle of $r=16$ with 8 equally spaced semicircles of radius=8. Each semicircle has one end on the larger circle's center and the other on the circumference of the larger ...
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185 views

Ratio of Radius of Circle $B$ to Radius Of Circle $A$ in the form $a + b\sqrt{c}$

The full question is as follows: Suppose $X, Y, Z$ are three different, circles of equal radius which are mutually tangent. Let circle $A$ be the circle tangent to $X, Y$, and $Z$ inside the gap ...
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What does the “Eternal Round” topological space look like?

I shall identify points on a circle with their angle in this question. For example, $0$ and $2\pi$ both correspond to the top point of the circle. Take the function $d(\theta)=2\theta$, which is a ...
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Prove that the circle contains the polygon.

Given a convex polygon. The circle is constructed for every triple of consecutive vertices of the polygon.We get the n circles. Select the circle with the largest radius. Prove that the circle ...
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68 views

concurrency of three lines on IO

Let $ABC$ be a triangle, and let $X$, $Y$, and $Z$ be the excenters opposite $A$, $B$, and $C$. The incircle of triangle $ABC$ touches $BC$, $CA$, $AB$ at points $D$, $E$, $F$, respectively. Finally, ...
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unit circle trigonometry where angle is greater than 90 degrees.

How is it possible to have sin of angle greater than 90 degrees. if sin is ratio of opposite side and hypotenuse in right angled triangle then triangle with one of the angle greater than 90 can not be ...
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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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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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A Problem With Coordinate Systems

Consider a coordinate system $\cal{C}$ such that the concentric half circles around two fixed points $P_1,P_2$ in the plane above line $P_1P_2$ create the grid. So any point in the upper half plane in ...
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Shortest closed Steiner chain of circles sandwiched between circumcircle and incircle of triangle

Consider Steiner chain of circles with the external circle of radius $R$, the internal circle of radius $r$ and $n$ circles in a chain with the radii $r_1,\dots,r_n$. Known condition for the ...
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Average height of a semicircle. Why are these different answers?

In my mind, there are two approaches to finding the average height of a semicircle. Integrating Cartesianally: $$y_{av}=\frac{1}{2r}\int_{-r}^r\sqrt{r^2-x^2}dx=\frac{\pi r}{4}$$ Or integrating polar:...
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Finding a metric to make a certain curve a circle

Talking with a friend of mine about a geometric problem, and, joking about a very bad drawn circle of him, he said: "That's a circle. Do not ask me the metric, but that's a circle". Such an ...
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If a tangent drawn from a point on the director circle of ellipse $E = \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, the chord…

Actually I am not getting the question, if the tangent is drawn from one point on the director circle of the ellipse itself then where is the chord of contact. Can you help me with a diagram???
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250 views

Homeomorphisms of the circle

I know that there is a vaste litterature about the group of the homeomorphisms of the circle. I would a good reference to start the study of this topic. Thanks in advance.
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How to decompose a 2d shape into sin and cosin modes?

Assume that you have a circle with radius $r_0$, then you keep adding cosine modes as below: $r=r_0+a_1\cos(1\theta)+a_2\cos(2\theta)+a_3\cos(3\theta)+a_4\cos(4\theta)+~...$ if you plot this as ...
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476 views

Why does (h,k) generally represent the center of a circle?

Why are h and k generally used to denote the coordinates of the center of a circle? After a bit of research, we found that h may represent "horizontal shift" or "horizontal translation", but we're ...
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105 views

Centroid and circumcenter — how close?

Suppose $R$ is some planar region, bounded by a curve. Let $C_1$ be the centroid of $R$, and let $C_2$ be the center of the "circumcircle" (the smallest circle enclosing $R$). Intuitively, it seems ...
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Prove special case of Brianchon's theorem using inversion

Brianchon's theorem says: When a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. From interactive demo: (...
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A question concerning radians and arc length

I was asked by a colleague yesterday about how the formula for the arc length of a circle is derived. I wanted to give them a correct answer, so I said I'd get back to them once I'd thought about it ...
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Problem about cyclic quadrilaterals

In cyclic quadrilateral ABCD, let E, F, G, H be the orthocenters of triangles BCD, CDA, DAB, ABC, respectively. Prove that EFGH is cyclic. Progress So far, found that if E is orthocenter of BCD ...
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Prove three chords of a circle are concurrent iff their poles with respect to a circle are collinear.

This probably would be a very simple problem if I could use any theorem I wanted about poles and polars, but in the book they give a definition and they say the problem should be solved using only ...
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How does this polar function behave?

I came across this question in my textbook for Nonlinear Optimisation and I don't know what to do: Consider the function: $$ f(x_1,x_2)=(r-1)^2-\frac{1}{2}(r-1)^2\cos \left( \frac{1}{r-1}-\phi \right)...
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Can you explain the solution of this geometric problem

A year ago IBM research posted an interesting geometrical problem: A gardener plants a tree on every integer lattice point, except the origin, inside a circle with a radius of $9801$. The trees are ...
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Ellipses touching a circle

Given a circle and two points $A$, $B$ in the plane, how do I find an ellipse with focal points $A$ and $B$ that touches the circle? How many such ellipses are there (at least/at most)? Can I ...
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Characterizations of a linear fractional transformation

Consider the function $$ g(t) = \frac{1+it}{1-it} = \frac{1-t^2}{1+t^2} + i \frac{2t}{1+t^2}. $$ (The second equality holds except when $t=i$.) It seems to be widely known that this function is the ...
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Topology of a 3D wired Mandala?

There is a so called 3D-wired Mandala, based upon $2$ large circles each flowered symmetrically on its circumference by two sets of each $8$ half-circles. The circles are interconnected together by ...
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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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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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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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Where does the square intersect with the circle?

A circle is drawn with center at origin, O, and radius $6$ cm. Find the coordinates of all intersections of the circle with an origin centered square of side length $10$ cm whose sides are parallel ...
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Find value of $\frac{\sum AA_1 \cos\left(\frac{A}{2}\right)}{\sum \sin A}$

Triangle $\Delta ABC$ is inscribed in a circle of radius one unit. If the internal angle bisectors of angles $\angle A, \angle B,\angle C$ meets the circle at the points $A_1,B_1,C_1$ respectively. ...
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A chord of length $6$ subtends an $80^\circ$ central angle in a circle. Can we calculate the distance from center to chord without trigonometry?

I know that the following can be answered easily using trigonometric ratios, but is there any way to go about it without relying on trigonometry? (The book from which the problem was taken doesn't ...
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Need help with a problem regarding Descartes' theorem

I'm working on a general Apollonian gasket (i.e., with no particular symmetry). One example might be to populate a Steiner chain with Apollonian circles. I programmed a recursive Descartes' theorem ...
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How to find the largest regular shape inside irregular convex polygons in 2D

I want to find a way that can find a regular or common shape (e.g a circle, triangle, square or rectangle) that has the maximum area inside different irregular 2D polygons. From this article, it ...
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In triangle $ABC$ show that…

Assume $ \triangle ABC$ is a right triangle where $\angle C= 90°$. Suppose that $ \overline {AX} $ bisects $\angle A$ and point $X$ lies on $BC$. Assume that the circumcircle of triangle $AXB$ ...
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Confusion in equation of pairs of tangents drawn to a circle

I have learnt that the equation of a tangent drawn ‘at’ a point $(x_1,y_1)$ to a circle $x^2 + y^2=a^2$ is $T=0$ that is $xx_1 + yy_1 - a^2=0$ where $T$ is obtained by replacing $x^2$ by $xx_1$ and $...
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Find width of the biggest rectangle in a sector

Is it possible to find the width and starting coordinates of the biggest horizontal rectangle of a fixed height inscribed in a given sector with any angle? This is actually to do with computer ...
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Finding equation of circle with the given restrictions

Tangents are drawn to circle $x^2+y^2-6x-4y-11=0$ from point $P(1,8)$ touching circle at $A$ and $B$. Let there be a circle whose radius passes through point of intersection of circles $x^2+y^2-2x-6y+...
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circle packing on a truncated hemisphere

I have a truncated hemisphere: z= $\sqrt{ R^2 - x^2 -y^2 } $ for $0 \le z \le (R -z_{top})$, where $z_{top} \approx R/3$ or $R/4$. I'd like to cover this truncated hemisphere with circles of radius $...
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Introduce a generalization of the Pappus theorem and the Pascal theorem

The theorem proved in A chain of six circles associated with a conic Let $1, 2, 3, 4, 5, 6$ be six arbitrary points in a hyperbola Let $1'$ be arbitrary point in the hyperbola. The circle $(121')$ ...
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3 “non-collinear” points create a unique circle using taxicab geometry?

I know that in standard two-dimensional Euclidean space three non-collinear points have a unique circle that touches all three points. I saw that @dan uznanski gave a determinant form for finding the ...
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Poles of tangents

Show that the poles of tangents of the circle $(x-p)^2 + y^2 = b^2$ with respect to the circle $x^2 + y^2 = a^2$ lie on the curve $(px-a^2)^2 = b^2(x^2+y^2)$. My attempt: Tangent to $(x−p)^2+y^2=b^2$...
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Is it possible to deform any given graph of a function into a circle?

Say that we have a circle in the $xy$ plane with radius $r$ and center $(C_x,C_y)$ and a point given by $(p_x,p_y)$. I was able to deduce that this point can be brought onto the surface of the circle ...
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How to calculate $\Delta$ in conic sections?

When learning conic section I learnt about $\Delta$. For any conic in general form : $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ Here $\Delta=abc +2fgh - af^2 - bg^2 -ch^2$ The conic is said to be ...