For questions conserning circles. A circle is a curve composed of points in a plane that are at a fixed distance from a fixed point.

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How to rotate a line segment around one of the end points?

I am given x1, y1, x2, y2 and θ. How can I find x3 and y3? By the way, there can be another line segment on the other side of AB (as if the line was rotated counter-clockwise). How to find that ...
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1answer
21 views

Calculating radius of circles which are a product of Tangent Intersections using a Regular Polygon

Introduction Lets have a regular polygon of $n$ sides inscribed in a circle of radius $H$, then construct tangents between the circle and each point of the polygon and draw new circle(s) trough the ...
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2answers
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If you colored every point of a circle 1 of 2 colors, is there always 2 same-colored points of distance $R$ apart?

If every point on a circle of radius $R$ in $\mathbb{R}^{2}$ were colored one of two colors, is there necessarily two points that are of the same color and of distance $R$ apart? what about $>2$ ...
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1answer
34 views

Can cyclic quadrilateral be a parallelogram $?$

Question: a) Can cyclic quadrilateral be a parallelogram $?$ b) Can a parallelogram be cyclic $?$ Solution: a) Cyclic quadrilateral can be a rectangle or square. They are also parallelogram. So ...
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2answers
2k views

Are similar circles really a thing?

I'm a fifteen year old who is currently studying circle geometry (if that is the appropriate term) and our teacher stated that concentric circles are similar. I thought about this, and it doesn't make ...
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1answer
28 views

Center of Circle given Apothem and 2 points

I am given 2 end points of the chord $AB$ as well as the apothem, the distance from the center point of the circle to the chord. I can easily find the radius circle and midpoint of the chord I'm ...
4
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2answers
86 views

Fit 2600 equally spaced points on concentric circles

My friend is working on an art project where she wants to draw 2600 dots on a circular table, symbolising the 2600 deaths of the conflict in east Ukraine. She approached me to solve this, but I've run ...
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3answers
113 views

Show that $O$ traces out a circle in the pencil defined by $A$ and $B$

Show that $O$ traces out a circle in the pencil defined by $A$ and $B$
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1answer
32 views

Finding the area of the region bounded by the incircle and the sides of the triangle?

click here for the image In an isoscles triangle, we can find the radius of the incircle by using the fact that the angle bisector of the third (unequal) angle is the perpendicular bisector of the ...
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2answers
85 views

If A, B, C, D are four points on a circle in order such that AB = CD, prove that AC = BD.

If A, B, C, D are four points on a circle in order such that AB = CD. How do you prove that AC = BD.
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0answers
47 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 ...
6
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1answer
64 views

Calculating radius of circles which are a product of Circle Intersections using Polygons

Lets say you imagine a circle with the radius $R$ and you inscribe a regular polygon with $n$ sides in it, whose side we know will then be: $$a=2R*sin(\frac{180}{n})$$ Then you draw a set of circles ...
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4answers
42 views

How far apart are centers of a set of Johnson Circles if the centers are equidistant?

I'm finding it hard to find the answer to this problem, I suspect it is simple and I'm missing something. Assume there are three circles with equal radius. The circumference of the circles intersect ...
5
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3answers
62 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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1answer
51 views

Three circles intersect at one point.

If three circles intersect at one point then there's unique $x$ and $y$ coordinate values such that the following equations are satisfied: $$(x-x_i)^2 + (y-y_i)^2 = r_i^2$$ Where $i=1,2,3$ Taking ...
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0answers
18 views

circle segment height by given fill fraction

At work I was facing the problem of how to calculate the height of a water column inside an horizontal cylinder given the volume of the liquid. A plot of this function and a visual explanation can be ...
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2answers
46 views

Bézier curve approximation of a circular Arc

I would like to know how I can get the coordinates of four control points of a Bézier curve that represents the best approximation of a circular arc, knowing the coordinates of three points of the ...
0
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1answer
37 views

length of a tangent

The two tangents to a circle are represented by $2x^2-3xy+y^2=0$ . A circle of radius=3 is in first quadrant . "A" is a point of tangency where one of these lines meet.What is length OA where $O$ is ...
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1answer
21 views

Formula for cycloid?

Is there a formula for cycloid? My approximation is $((2\times(x\div(\pi\div2)))-(x\div(\pi\div2))^2)^.626$.
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0answers
33 views

Find center and radius of circle with n# of equally spaced points

Say there is a point P, with coordinates $(x_1,y_1)$, and there is a circle that passes through this point, and the origin. There are n# of equally spaced points that lie on the circle leading from ...
0
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1answer
39 views

Finding the area of a part of two internally touching circles [duplicate]

Two circles touch each other internally at point A as shown in the figure: (http://imgur.com/hmzgMCT) O is the centre of bigger circle. If CB = 9 cm and DE = 5 cm. Find the area of the crescent ...
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4answers
78 views

Semicircular paper and creasing of a chord

A semicircular piece of paper with radius $2$ $cm$ is folded along a chord so that the arc is tangent to the diameter.If the contact point of the arc divides the diameter in the ratio $3:1$,determine ...
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1answer
50 views

How to adapt “System of Circles” method to 3D for finding a sphere given 4 points?

I want to analyze (computational complexity & running time) of different approaches to determining a sphere in 3D given 4 points on its surface. To start I have been searching for different ...
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1answer
51 views

Find the coordinates of the points where the two circles intersect. [duplicate]

The two circles are: 1) $$(x-2)^2 + (y+1)^2 = 25$$ 2) $$(y-2)^2 + (x+1)^2 = 25$$
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0answers
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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, ...
6
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1answer
165 views

Construct a triangle with its orthocenter and circumcenter on its incircle.

Construct $\triangle ABC$ such that its orthocenter ($H$) and circumcenter ($O$) are on its incircle. I've tried something by inverting everything WRT circumcircle but don't have proper idea... ...
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4answers
79 views

Different way of finding area of a circle. [duplicate]

Hello maths community! One day I was solving a geometry problem and I thought I had found a way of solving it. When I was solving the problem, I kind of invented a new way of finding an area of a ...
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2answers
44 views

How to find the equation of circle that passes through ($5,3$) , ($7,-2$) and ($-4,4$) circle with center at origin ($0,0$) and radius $r$?

It is a challenge assignment on our class and I can't figure out how to solve it I always got stuck it is not the same as the other examples which are easy to solve. thanks
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2answers
19 views

Finding the Equations of a Circle provided a point and the radius

So I tried googling the exact question and I never found the solution. This is homework so I really don't want to know the answer but how to arrive at the answer. The question that was given was ...
2
votes
2answers
31 views

Family of circles with $AB$ as diameter

The circle $S_1 :x^2+y^2-4=0$ cuts the circle $S_2 :x^2+y^2+2x+3y-5=0$ in A and B. Then find the equation of circle with $AB$ as diameter. Answer is $13(x^2+y^2)-4x-6y-50=0$ Equation of AB will ...
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1answer
23 views

Work out radius from arc sector and angle

I am trying to work out the radius of a sector with arc length of 47.6 and a angle of 210. I tried a formula which was $r=\frac{L}{2π}\times \frac{360}{\theta}$ I saw but kept ending up with 128.176 ...
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2answers
50 views

Calculate the diameter of an inscribed circle inside a sector of circle [closed]

$AOB$ is a sector of a circle with center $O$, angle = 45° and radius $OA=10$. Find the radius of the chord inscribed circle in this sector such that it touches radius $OA$, radius $OB$ and arc $AB$. ...
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1answer
62 views

Common tangent to two circles

I need to find the common tangents to the following circles: $~~~~~~~~\mathscr C_1:~x^2+y^2+8x+2y-8=0~~~~~$ and $~~~~~\mathscr C_2:~x^2+y^2-16x-8y-64=0$ How could I go about this? I found a ...
0
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1answer
34 views

Conics and Loci Question (Hyperbolae and Circles)

A circle has the equation $x^2 + y^2 = r^2$. Tangents are drawn from a point $P(x_1,y_1)$ to the circle and these touch the circle at points $A$ and $B$. If the position of $P$ can vary and the locus ...
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0answers
37 views

Question based on circles?

In a circle, ABCD is a cyclic quadrilateral and PQ is a diameter. PQ intersects the side AD and BC. Prove that QC and AP bisect Angle C and Angle A respectively.
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1answer
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What is wrong with this reasoning when calculating circle perimeter? [duplicate]

Looking at the following image, which was posted on the internet: Could someone tell me what is wrong? It seems true for the first 4 small images. But, when it comes to infinitesimal length, ...
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1answer
41 views

Side of a square

How to find the side length of this type of squares if $A$ and $B$ are given
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1answer
45 views

Circle Problem of Chords [closed]

Please tell me what and how this equation works.
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1answer
52 views

Hard Integral from a special case of Fourier series

I was trying to find the equation of a sinusoidal function that has one upward facing semicircle and immediately after one downwards facing semicircle which would make up a period of the function. ...
2
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0answers
41 views

A circle can include all but one of n points, but which one can it be?

The answers to the question "Circle enclosing all but one of n points" demonstrate that, given $n$ points, it is possible to construct a circle such that all but one of the points is inside the circle ...
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1answer
171 views

Quadrilateral with maximum area inside a semicircle [closed]

The quadrilateral $ABCD$ inside a semicircle with radius 1 has a maximum area. Calculate that maximum area. Clues to solution include the observation that the vertices of such quadrilateral must be ...
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1answer
50 views

Geometric interpretation of Leibniz formula for $\pi$

We know $\pi=4(\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}....)$. I'm wondering, is there a geometric interpretation of this identity. Can we prove this identity by finding a different way to ...
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0answers
35 views

The intouch triangle and excentral triangle

Let $I_a$, $I_b$, and $I_c$ be the excenters of triangle $ABC$, and let $D$, $E$, $F$ be the intersection points of the incircle to segments $BC$, $AC$, and $AB$. Prove that the center of homothety ...
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0answers
10 views

Close-packed spheres diameter of circle that can fit between spheres

Imagine small spheres placed in their most efficient arrangement (close-packed). Sort of like if you round up a lot of billiard into a corner, you can imagine the pattern they would form. I would like ...
0
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1answer
53 views

Equation to the circle.

How to show that the equation to the circle of which the points $(x_1,y_1)$ and $(x_2,y_2)$ are the ends of a cord of a segment containing an angle $\theta$ is, $$(x-x_1)(x-x_2)+(y-y_1)(y-y_2) ± ...
2
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4answers
43 views

Problem based on circle geometry - related to circumcircles and angles finding the angles within a circle

Let the vertex of an angle $ABC$ be located outside a circle and let the sides of the angle intersect equal chords $AD$ and $CE$ with the circle. Prove that the angle $ABC$ is equal to the half the ...
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1answer
58 views

Two tangents $BC$ and $BD$ are drawn. Prove that $OB=2BC$

Two tangent segments $BC$ & $BD$ are drawn to a circle with centre $O$ such that $\angle{CBD}=120^{\circ}$. Prove that $OB=2BC$. What I've tried, $BC=BD$[two tangents drawn from a single point ...
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2answers
42 views

Algorithm to cover maximal number of points with one circle of given radius

we have a plane with some points on it. We know coordinate of each point apriori. We also have a circle of unit radius. I need an algorithm that determines optimal/sub-optimal position of a circle ...
0
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3answers
41 views

How to make a semicircle graph?

What is the formula to make a semicircle graph that is continuous? By continuous I mean like a sine or cos graph but shaped like semicircles one after the other. Thanks
0
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1answer
41 views

Is the Locus circle?

Locus of points such that sum of it's distances of them from four fixed points remains constant? Is the locus circle? I was not able to solve it as there were four radicals. Is it a theorem?