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.

692 questions with no upvoted or accepted answers
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9
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1answer
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Relationship between two centers of circles in a Venn diagram

Let $S$ be a circle of 1 unit area. No part of circles $A$ and $B$ are outside the circle $S$. Let $n(S)=1$, $n(A)$, and $n(B)$ be the area of circle $S$, $A$, and $B$, respectively. For the given ...
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2answers
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New Proof of Pythagorean Theorem (using inscribed circle)?

I was solving an easy problem for fun when I stumbled onto this, and was wondering if this was a correct and possibly a new proof of the Pythagorean Theorem. Given right triangle $\triangle ABC$, and ...
7
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412 views

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 ...
6
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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 ...
6
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179 views

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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160 views

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 ...
6
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1answer
2k views

The number of circles that will fit inside the area of larger circle?

Let's say circle $\omega_1$ has a diameter $X$. Let $X>Y$; $Y\in \mathbf{R}^{+}$. How many circles with diameter $Y$ will fit inside $\omega_1$? Is there a formula for this?
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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 ...
5
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3answers
100 views

Right Triangle and Circle Theorem

Let $ABC$ be a triangle such that $\angle BAC$ is a right angle. Suppose $D$ is a point lying on $BC$ such that $BD=1$, $DC =3$ and $\angle ADB=60^{\circ}$, find the length of $AC$. I was told that ...
5
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768 views

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 . ...
5
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1answer
84 views

Length from tangent circles

A circle $Γ_1$ of radius $25$ is externally tangent to a circle $Γ_2$ of radius $16$ at $C$. Let $AB$ be a common direct tangent, so that $A$ lies on $Γ_1$ and $B$ lies on $Γ_2$. Draw the tangent to $...
5
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1answer
100 views

explaining the resriction $b<a<2b$ in a triangle

I saw in a book that if $ABC$ is an isosceles triangle $(AB=AC)$ and the triangle is tangent to a circle in points $D,C$ and $AC$ is intersecting the circle in point $E$; $AC=a$, $BC=b$ so it has the ...
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828 views

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 ...
4
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1answer
107 views

The circumference of the circle is $C$, what is the area of circle in terms of $C$?

The circumference of the circle is $C$, what is the area of circle in terms of $C$? a). $\dfrac {C^2}{4\pi }$ b). $2\pi C$ c). $\dfrac {4}{3} \pi C^2$ d). $2\pi C^2$ My Attempt: $$\textrm {...
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187 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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83 views

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 ...
4
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1answer
2k views

Find the equation of the circle whose diameter is a chord.

$Y=mx$ is a chord of circle of radius $a$ through the origin whose diameter is along the $x$-axis. Find the equation of the circle whose diameter is the chord. We also need to find the locus of ...
4
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0answers
110 views

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 ...
4
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3answers
265 views

Is $\sin(x)$ =$-\sin(180^o+x)$?

I figured out that $\sin(x)$ should equal $-\sin(180+x)$ like in this picture But when I type on Wolfram $$\sin(a\mathrm{deg})=-\sin(180+a \mathrm{deg})$$ it says it's false. Why? I've tested it ...
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69 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, ...
4
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1answer
332 views

Longest chord inside the intersection area of three circles

I am currently working on my masters thesis in computer science and I stumbled onto a geometry problem. My goal is to compute the length of the longest possible chord inside the intersection area of ...
4
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1answer
541 views

Find circles that completely cover a polygon minimizing the amount of space covered outside the polygon

I have an arbitrary polygon that I need to roughly represent using circles. Any point inside the polygon must lie inside a circle. There will be points outside the polygon that will fall under a ...
4
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1answer
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Find normal vector of circle in 3D space given circle size and a single perspective

I don't really know what to search up to answer my question. I tried such things as "ellipse matching" and "3d circle orientation" (and others) but I can't really find much. But anyways... I have ...
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0answers
453 views

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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38 views

Proof verification: the angle subtended by a chord can never be 90 degrees

I couldn't find any sources of this online, so I would like to ask if what I'm proposing below is correct, or if a similar theorem has been proven before. We know that the angle subtended by the ...
3
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1answer
86 views

Maximum total area of n non-intersect circles?

Given n points on the x-axis, we give arbitrary radius for each point such that each constructed circle doesn't overlap another constructed circle from another point. Which means these circles do not ...
3
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0answers
29 views

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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38 views

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 ...
3
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1answer
39 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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0answers
69 views

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 ...
3
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0answers
55 views

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 ...
3
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113 views

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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0answers
72 views

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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0answers
38 views

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???
3
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1answer
73 views

Given distances from an external point to 2 points on a circle and its center, what is the radius of the circle?

From a point outside a circle, I know the distance to two (or more if needed) points along the circle, as well as the distance to the center of the circle. Angles are unknown. How can I calculate ...
3
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0answers
255 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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0answers
111 views

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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0answers
484 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 ...
3
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1answer
2k views

Relationship between incenter and circumcenter

Let ABC be an acute triangle with circumcenter O and incenter I. Points E, M lie on AC and F, N on AB so that BE ⊥ AC, CF ⊥ AB, ∠ABM = ∠CBM and ∠ACN = ∠BCN. Prove that I lies on EF if and only if O ...
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106 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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0answers
192 views

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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0answers
103 views

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 ...
3
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0answers
122 views

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 ...
3
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0answers
303 views

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 ...
3
votes
1answer
156 views

Circle with perpendicular chords

A blue circle is divided into $100$ arcs by $100$ red points such that the lengths of the arcs are the positive integers from $1$ to $100$ in an arbitrary order. Prove that there exists two ...
3
votes
1answer
490 views

Find Area of 3 Sector Circle, Variable center point

I have a Circle separated into 3 sectors. At start each sector has the same central angle, 120°. Therefore each sector should be taking up the same area. I want to be able to move the center point ...
3
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0answers
31 views

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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213 views

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 ...
3
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0answers
487 views

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 ...
3
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0answers
114 views

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 ...