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Questions tagged [circle]

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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The good pseudo-half-circle-like curves that quadratic bezier curves can create

Since quadratic bezier curves can only approximate circles, I'm wondering what they can actually do well. That is, the sort of half-circle-like shapes that quadratic curves can make. For instance, in ...
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I want to determine the lenght of $x$ by making equations for each triangle ACD and ADB.

I want to determine the lenght of $x$ by making equations for each triangle ACD and ADB. Let's recall $\angle DAB = \alpha $, $\angle DBA = \beta $ in triangle ADB $$\alpha + \beta + 90 = 180$$ $$\...
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Generalizing computation of number of pixels in inscribed circle to axis-aligned ellipses at arbitrary points

This answer really nicely sums up the question of how to compute the number of pixels inside an inscribed circle. However, I am looking for a more generalized version of this in two ways. 1) I would ...
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Ratio of axes in an approximate circle

I have some shape that is approximately circular and has area = 100 pixels, with every pixel having area 1. Is there some mathematical way I can define the ratio of the longest axis to the smallest ...
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What is the value of $a$ where $√a$ is area of a trapezoid which touches the circle with center $O$ (diameter is 2)?

The sides $AB, BC, CD$ of trapezoid $ABCD$ touches the circle with center $O$ and they are equal. $AD$, goes through the point O. If diameter is 2, then the area of the trapezoid is $√a$ . What is ...
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Inscribed circles radius

Let ABCD parallelogram. The inscribed circle in triangle ABD is tangent to BD in E. Show that $$\frac{r_{DEC}}{r_{BEC}}=\tan (\frac{1}{2}\angle ACD)\cdot \tan (\frac{1}{2} \angle ADB)$$ What I have ...
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A query on a cyclic pentagon

Let ABCDE be a cyclic pentagon, where AC=2, AD=3, BD=5, BE=1, CD:DE=10:3(Proportion mark, division sign isn't available in my keyboard). What is the value of BC:CE ? I worked with the area of ...
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41 views

Finding the diametre of two circle with the given condition

Two circle as shown in the figure, A is the tangent point of both the circle. B is the centre of the large circle. The distance of CD = 90 mm(according to estimation) and EF = 50 mm. What is the value ...
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1answer
28 views

Perimeter of a shaded part in a circle

https://cdn.discordapp.com/attachments/334723040099434498/536393147538997250/Screenshot_20190120-055431.jpg This is a question that i want to know will the solution of it be 8 pi or 8 pi + 12 , will ...
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Why don't circles have infinite degrees? [duplicate]

A triangle has 180 degrees. A square has 360. A pentagon has 540. A hexagon has 720. An octagon (which is starting to look a lot like a circle) has 1080. You see the trend?
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Side of the equilateral triangle

I tried very much but since tomorrow is my exam, i cannot risk it. The following is a geometry problem, which i have tried very much but could not grasp a solution. I think that i require pythagoras'...
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1answer
64 views

Circle rotating in circle rotating in circle

There are three circles with different radii arranged such that the smallest circle is contained entirely within the next larger circle and that circle is entirely contained within the largest circle. ...
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1answer
37 views

Area of a circle segment on sphere, given radius (meters) and central angle (degrees)

Situation I have a circle segment and some information about the circle it belongs to. Given Information: radius of the circle in meters central angle in degrees lat/long of all three points on the ...
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How to double the circle?

I'm looking for a compass-and-straightedge method to construct a circle that has area twice of the area of another circle, with no prior knowledge of π, without knowledge of the formula for the area ...
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1answer
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Condition for which $|z-z_1|^2+|z-z_2|^2=K\in\mathbb{R}$ represent a circle

Let $K\in\mathbb{R}$ and $z_1,z_2\in\mathbb{C}$. Prove that the equation $|z-z_1|^2+|z-z_2|^2=K$ represent a circle iff $K\geq\frac{1}{2}|z_1-z_2|^2$. My Attempt $$ |z|^2+|z_1|^2-2\mathcal{Re}(...
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Volume of revolution of a triangle.

I have a triangle on the $xy$ plane whos base's center is $x_1$ away from the $y$-axis horizontally. I want to rotate it around the y-axis and find the volume of the triangular doughnut shape. My ...
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Two circles X and Y with centres A and B intersect at C and D. If area of circle X is 4 times area of circle Y, then AB=?

This question is solved by taking angleACB = 90 in my book. How can we say that this angle a right angle triangle? Given answer is √5r.
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if triangle ABC has incenter I how is it possible to show the circumcenter of triangle BIC lies on the circumcircle of triangle ABC?

if triangle ABC has incenter I how is it possible to show the circumcenter of triangle BIC lies on the circumcircle of triangle ABC? D is the circumcenter of BIC
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Optimization exercise regarding a circle and a function

A circle $\omega$ centered at $K$, $(0.5,0.5)$ is tangent to the $x$- and $y$-axes. Consider the function $$f(x):=\frac{8}{4+x^2} \; \forall x\in\mathbb{R}$$ and a point $P\in f$ such that the ...
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Calculate center coordinates of circles surrounding a larger circle

I want to draw, say, 8 smaller circles that are adjacent to the big circle the edge of a big circle, similar to this picture. I know the center coordinates of the bigger circle $(A, B)$, its radius $(...
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Circle circumference point calculation

In the image below, I have a part of a circle. Given, $$\text{chord }d=1050\ mm\\ \text{height }f=50\ mm\\ \text{radius }R=2781\ mm\\ \text{centre }O(700\ mm,2781\ mm)$$ and $3$ points $A(0,200), B(...
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2answers
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Draw a multipolygon

I need to build a shutter like this: What is the formulas to draw each vertex? Suppose I have num which is the number of triangles (in this case ...
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how to find circle circumference point from a point inside the circle

given the image bellow how can i find the points of a circles circumference from a point inside the circle with given X and Y ? also can how to calculate it from a point outside the cirlce ? https://...
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Understanding Bezier Curves vs. Circular/Elliptical/Other Arcs

From what I've read, you cannot construct an elliptical or circular arc with a single bezier curve (though I read maybe you can if the arc is less than 1/4 of a circle or something small like that, or ...
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Prove that a point in a circle equidistant from any three points on the circle is the centre

Prove that a point in a circle equidistant from any three points on the circle is the centre $\frac{1-1}{1-1}=?$
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Calculate Overlapping Area of $2$-Dimensional Shapes

I am running a Computer Simulation where 2 Shapes are moving towards each other and will eventually overlap. I want to calculate the overlapping Area of the shapes - in this example a Circle and a ...
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How to find dimensions of partial circle (“arc”) of a given angle

I tried to draw this to make it hopefully more straightforward to understand. So here is an angle drawn that demonstrates a small rounded corner: The bounds of the rounded corner can be drawn like ...
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1answer
44 views

General way of modeling Bézier curves and circles

So it turns out that you can't totally model circles with Bézier curves: How to create circle with Bézier curves? I'm wondering if there is a mathematical system or construction that unifies circles,...
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1answer
36 views

Relating Turning Angle To Arc

I am trying to determine how to relate the turning angle of a vehicle to an arc. For the purposes of this vehicle, I am not taking into account Ackerman Steering, so for the intent of this question, ...
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43 views

Chord of a Circle [closed]

Simple question regarding chords in a circle. Is the midpoint of a circle's chord always perpendicular to the circle's centre?
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Proving 2 chords are the same length in a circle divided into $n$ equal arcs

Let us have a circle that is divided in to $n$ equal arcs by $n$ points on the circumference. There are $\dfrac{n}{2}$ chords joining pairs of points. For what values of $n$ would it be necessary for ...
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What do “3 different points” have to do with linear dependence in determining a unique circle?

In learning about the general formula for a circle: $$(x-h)^2 + (y-k)^2 = r^2$$ my book states that $3$ points are sufficient to guarantee the solution (or absence of solution) for an unique circle. ...
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Find all integer solutions of $3(m^2 + n^2) - 7(m+n) = -4$.

The solution and further information about how to solve this type of equation about how to solve this type of lattice point and circles will be much appreciated. Thanks in advance.
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In $\triangle ABC$ with $D$ on $\overline{AC}$, if $\angle CBD=2\angle ABD$ and the circumcenter lies on $\overline{BC}$, then $AD/DC\neq 1/2$

Let $\alpha$ be $\measuredangle ABD$ Let $\beta$ be $\measuredangle DBC$ Let D be a point on AC such that BD passes through the origin point O Prove that $\frac{AD}{DC}$ cannot be equal to ...
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60 views

Show that $CD\perp AB $.

Let $\triangle ABC $ with all angles $<90°$. Let $A_1$ the middle of $ [BC] $. If $\angle BAA_1=30°$ and $D\in [AB] $ s.t. $CD=AB $ show that $CD\perp AB $. My idea : I draw $A_1T\perp AB $, $T\...
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Show that $AB=AA_1$

Let $ABC $ a triangle with $ A_1$ the middle of the edge $[BC] $ and $BAA_1=30°$. $ Let D\in [AB] $ s.t. $CD=AB $. I have to show that $AB=AA_1$. This conclusion seems to be wrong because I can ...
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circle sliding along the inside of another circle is a straight line

Note: Not a duplicate of Tracing the path of the point as I have my own method of arriving at the answer which seems wrong I saw this video in which they show that the trajectory of a point on a ...
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Calculate new velocity of a ball when colliding with a circle in 2D

I'm currently making a 2D game in C++ but i've come across a problem while doing collisions. I have a ball free falling The above example shows what's going on in my game, I can detect when the two ...
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103 views

Circle whose radius is infinite

I have the intuition that a circle whose radius is infinite is a straight line. Nonetheless, I don’t feel that what I’ve just stated is really scientific as it has some vagueness and lacks precision. ...
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1answer
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Does the equality of the sum of opposite sides in a quadrilateral necessarily imply the existence of an inscribed circle?

I had come across a question in which we had to show that a given quadrilateral, if subjected under a condition, had an incircle. So, will it be sufficient to show that $a+b=c+d$, if $a,b,c,d$ are the ...
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52 views

Circle transformation

Let $K$ be a circle $(x-1)^2+(y-4)^2=5$ and $P$ a line with the equation $y=-3x$. We transform the circle $K$ with a transformation, defined with the next conditions: points are tranformed, so that ...
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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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Examining whether a circle goes through any of a given set of points

In a rectangular coordinate system, the circle is created at the central and passes through point $P(0,-3)$. Which of the following points does it also pass through? $(3,3)$ $(-2 \sqrt 2,-1)$ $(2,6)$ ...
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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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If $S, S_1, S_2$ be the circles of radii 5,3 and 2 respectively. If $S_1$ and $S_2$ touch externally and they touch internally with $S$. [closed]

If $S, S_1, S_2$ be the circles of radii 5,3 and 2 respectively. If $S_1$ and $S_2$ touch externally and they touch internally with $S$. The radius of circle $S_3$ which touches externally with $S_1$ ...
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Why we don't get two tangents in the point form of tangent from a given point to a circle?

In my textbook I have read the point form of representation of tangent from a point $P(x_1,y_1)$ to a circle $x^2 + y^2 + 2gx + 2fy + c = 0$ which is given by $$xx_1 + yy_1 + g(x+x_1) + f(y+y_1)+c=0$$...
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Calculate the intersection of a line segment on the radius of a circle

Given the length and one endpoint of the line segment, how can we find the other endpoint so that it is on the radius of a circle (known coordinates and radius)? Assume that there is at least one ...
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How to find X valie of a point on an arc?

I have an arc with a start point (Xs, Ys) and an end point (Xe, Ye). Also, I know the direction of movement between them, the ...
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1answer
77 views

The value of $\alpha+\beta+\gamma$ is ______.

A variable line $ax+by+c=0$ where a, b, c are in AP, is normal to a circle $(x-\alpha)^2+(y-\beta)^2=\gamma$, which is orthogonal to the circle $x^2+y^2-4x-4y-1=0$. The value of $\alpha+\beta+\gamma$ ...
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87 views

Which type of triangle has the largest circularity value, $\sqrt{\frac{4\pi\cdot\text{area}}{\text{perimeter}^2}}$?

Equation 7 of "Particle Shape Factors and Their Use in Image Analysis–Part 1: Theory" (PDF) defines circularity as $$\text{circularity} = \sqrt{\frac{4\pi\cdot\text{area}}{\text{perimeter}^2}}$$ ...