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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How do you find the coordinate $(x,y)$ of Point $B$ given coordinates of $A$, the radius, and the angle within a unit circle?

Given coordinate positions of $A (x_1,y_1)$ on a unit circle. How do you calculate $B (x_2,y_2)$. The angle theta is known and radius R. See the below picture.
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(geometry) How to construct inscribed circle between 3 circles?

Suppose I have 3 circles not overlapping (but possibly touching) each other. Is there always an inscribed circle that is touching (i.e. tangent to) each of the 3 circles? And if yes, how do I ...
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A rigid bar suspended from $2$ wires, given the coordinates of the centre of the bar, find the lengths of the two wires

I'm not sure whether this is a geometry or mechanics problem, and I'm not too hot on either so either way I need help :) I think it's pure geometry. Also I am not really sure how to formulate this ...
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2answers
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How do you convert $(x-r)^2 + y^2 = r^2$ to polar form? Why do I get $0 = 0$?

I am trying to convert this circle from cartesian to polar form. I have googled and searched for various guides/youtube videos and followed them. The answer is supposed to be in polar form $r = 2 \cos ...
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1answer
30 views

Circumcentre coordinates [closed]

Let P be a moving point. G is the circumcentre of triangle PQR. The coordinates of Q, R and G are (6,9),(a,11) and (h,3) respectively, where h > 0. question(a): how to express the coordinates of G ...
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3answers
204 views

Calculate the distance of any point on the arc from the center of circle

This is my first time posting so I hope my formatting is correct. Consider this, I have two circles, one big one small with radius $r_1$ and $r_2$. The borders of both circles are touching. See image: ...
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1answer
50 views

What is the ratio of the area of a circle to its radius? [closed]

I solved this. $$πr^2/r=π r$$ In my book, they said it can not be determined. Am I right or wrong?
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Circle geometry

I met an interesting task, tried a lot of ways to prove but didn't find any. Problem: Let $KLM$ be a triangle, let $I$ be an incircle's center. Let $Q$ be the point on an extension of the segment $KM$ ...
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2answers
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Using vectors to solve geometric inequality questions

The question is : A triangle is inscribed inside a circle of radius $1$. The maximum value of sum of squares of the sides of triangle is? In the solution given for this question they have given to use ...
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1answer
107 views

Define polar coordinates of circle at origin and circle with radius $R$.

Question: (i) Define in polar coordinates $r = f(\alpha)$ the origin-centred circle with radius $R$. Specify the domain range for the polar coordinate $\alpha$. (ii) Define in polar coordinates $r = ...
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Inscribed Circles in a Quadrilateral

I found this problem online, where it was asked to prove EF = GH. I was able to prove that, but got intrigued by how the four smaller inscribed circles could be constructed in the first place. That ...
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Let $AD\cap (BFC) $ in points $P$ and $Q$ and let $AD\cap (ABE)=M$ then $MP=MQ$.

Let $\triangle DEF$ be the medial triangle of $\triangle ABC$ with standaring notations. Let $AD\cap (BFC) $ in points $P$ and $Q$ and let $AD\cap (ABE)=M$ then $MP=MQ$. Here is the diagram: There ...
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2answers
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How to prove the curves $s_1(\theta)=\{\sin\frac\theta2,\cos\frac\theta2\}$ and $s_2(\theta)=\{-\sin\frac\theta2,-\cos\frac\theta2\}$ form a circle

Say I have a set of two surfaces: $$s_1\left(\theta\right)=\{\sin\frac{\theta}{2},\cos\frac{\theta}{2}\}$$ $$s_2\left(\theta\right)=\{-\sin\frac{\theta}{2},-\cos\frac{\theta}{2}\}$$ where $0\leq\theta\...
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2answers
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Are $C$, $D$ and $E$ collinear?

Sorry for the picture. I really could not draw this in Geogebra. I want to find the length of $AC$. I assumed that $C$, $D$ and $E$ are collinear and solved it trivially using similar triangles. ...
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Is there a way to find the “roundness” of random points on a 2D plane?

I have multiple sets of random points and I want to find the one that is most "round" My thoughts are that if I were trying to find which set matched a particular function then the best set ...
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New conjugate ?!

I've been playing with amazing Geogebra software and finally found a construction that perfectly fits the description of a conjugate. Apparently there is a plethora of remarkable properties associated ...
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2answers
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How to deal with approximations involving “large” numbers

I came across this question: Here I am given a circle, with center O. 2 points $S_1$ and $S_2$ are present on the diameter of the circle at equal distances from the center. $P$ is a point on the ...
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Circle touching a given line at a point

The line $x=y$ touches a circle at a point $(1,1).$ If the circle also passes through the point $(1,-3)$ then find it. Circle touching a line is $$(x-x_1)^2+(y-y_1)^2+λL=0$$ we get $(x-x_1)^2+(y-y_1)^...
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1answer
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Tangent circles (distance)

You have two identical circles, with radius 5, on a horizontal plane standing precisely next to each other(so that the two circles are tangent to each other). Question1: On top of them a smaller ...
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Why is the angle at the centre is twice the angle at the circumference [closed]

I want to think logically and intuitively that why is the angle at the centre made by a cord is twice the angle at the circumference
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1answer
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How can I find the formula for a cycloid with a given speed?

I know that for a cycloid with radius $R$ and time $t$, it can be defined as $x = R(t - \sin t)$ and $y = R(1 - \cos t)$. However what if it's not at unit speed and we have a speed $z$/sec such that ...
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1answer
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Find the area of $\triangle BFC$ [closed]

Square $ABCD$ is inscribed in a circle. An equilateral $\triangle BEF$ is also inscribed in the circle. $F$ and $C$ lie on same side of $BD$. If radius of circle is $12$, what is the area of $\...
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Packing parabola with the chain of internally tangent circles

Related to the question Condition for perfect packing of ellipse with circles along the major axis An attempt of "perfectly pack" a parabola $x=ay^2$ as a special king of ellipse with ...
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1answer
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Why do circles (of the same sized circumference) not tesselate?

If our test for whether a regular polygon can tesselate with itself is whether the degrees of an individual interior angle can divide 360 to yield an integer (some examples of these integers are 6 [...
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3answers
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(BAMO $2013/3$) $ABH$, $BCH$ and $CAH$ is congruent to $ABC$.

Let $H$ be the orthocenter of an acute triangle $ABC$. Prove that the triangle formed by the circumcenters of triangles $ABH$, $BCH$ and $CAH$ is congruent to $ABC$. I have already seen many answers ...
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Show these three circles share their external common tangent lines

it is a silly problem but I would like to see some ways to show that the external tangent lines from $c_1$ and $c_3$ are also the external tangent lines from $c_1$ and $c_2$: $c_1$ centered at $A$ ...
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Find radius of circle for given chord length and circular segment area

I'm struggling with finding a circle radius $(r)$ of circular segment which has given chord lenght $(s)$ and circular segment area $(A)$. I'm interested only in solution when segment angle $(\alpha)$ ...
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1answer
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Constructing a tangent circle of fixed radius inscribed in a lens?

Sorry if this question is a tad poorly written, as I don't have much of a mathematical background and am actually a costume designer who's in a bit of a pinch. I have been trying to create an eye ...
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1answer
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Distance between vertex and orthocenter

This is a problem from an italian Olympiad contest: I don't need a solution of this problem, but I'd appreciate some hints, because all my attempts of solution have failed. Consider a triangle $ABC,$ ...
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2answers
248 views

Geometric proof of chain rule with the derivative of $\sin(2x)$

I'm following this post https://math.stackexchange.com/a/2169/612996 as my example and I've figured out how it works for $\sin(\theta)$, During my first try: I keep on missing the factor of $2$ when ...
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5answers
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Why point is a circle with radius zero?

I was reading this What is a point circle, a real circle and an imaginary circle? and i get confused with the statement that is written in the accepted answer , i.e A point "circle" is just ...
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1answer
43 views

Finding the area of two shapes

A circle of radius $5$cm and a square of sides $8$cm have the same centre. Find the total area of the region that belongs to either the square or the circle, but not both. I have constructed a ...
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1answer
51 views

Find the geometric locus of the intersecting lines connecting the endpoints of two chords.

The problem is from Kiselev's Geometry Book I. Exercise 594: Find the geometric locus of the points of intersection of two chords $AC$ and $BD$ of a given circle, where $AB$ is a fixed chord of this ...
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2answers
179 views

Given the following tangent circles chain to draw the next circle of the chain

the chain is from the yellow circles. What I want is a way to draw the next yellow circle given all the ones before it: I know how to draw the first circle $(P_0Q_0X)$. I know the (nameless in the ...
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3answers
61 views

If $P$ is on the circumcircle of a triangle, show that the feet of the perpendiculars from $P$ to the side-lines of the triangle are collienar [closed]

Let $ABC$ be a triangle and $P$ be any point on its circumcircle. Let $X,Y,Z$ be the feet of the perpendiculars from $P$ onto lines $BC, CA$ and $AB$. Prove that points $X,Y,Z$ are collinear. So I've ...
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How to build a function to “unsquash” a projection of one circle onto an axis.

Consider a function $f$ that takes an angle $\alpha$ in radians and $f(\alpha)$ computes the $2D$ coordinates that describe the shape of a circle. The function is defined on the interval $[0, 2\pi]$ ...
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1answer
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In Johnson circles, three lines each passing through the center of one of the circles and the $2$-wise intersection are concurrent.

The problem is from Kiselev's Geometry Book I. Exercise 591: Three congruent circles intersect at one point. Prove that the three lines, each passing through the center of one of the circles and the ...
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1answer
53 views

Circles and enneagon

Using that in a triangle ABC, $\tan\frac A2=\frac{r}{p-a}$ where $p=\frac{a+b+c}{2}$, I found that the radius are equal if $\tan^220°=\frac{\frac{1+2\cos40°}{\cos20°-1}}{1-\frac{1}{2\cos40°}+\frac{\...
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2answers
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The bisector of the exterior angle at vertex C of triangle ABC intersects the circumscribed circle at point D. Prove that AD=BD

The bisector of the exterior angle at vertex $C$ of triangle $ABC$ intersects the circumscribed circle at point $D$. Prove that $AD=BD$. So what I'm wondering is how to prove this? I've already drawn ...
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1answer
28 views

Find coordinates of parallel lines within a circle sector [closed]

Geometry is not my strongest subject and I can't find solution to my problem. I'm writing algorithm that finds coordinates in a circle sector. There is $n$ amount of parallel lines, each with equal ...
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3answers
79 views

Given two circles externaly tangent to each other and the common tangent line. Draw a third tangent circle.

We are given $\Gamma_A$ centered at $A$ and $\Gamma_B$ centered at $B$ tangent to each other externally at $C$. Line $DE$ is one common tangent to both not through $C$. Is there a nice way to draw ...
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0answers
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Finding the initial conditions for an incircle related vector problem

Question: Suppose there is a triangle with angles $\alpha$, $\beta$ and $\gamma$. The incircle to this triangle is drawn. A set of three vectors is defined such that they each start from a distinct ...
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3answers
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How do I create an offset shape that is a specific distance from a given circle, in the direction of the origin?

I'm an amateur engineer, working on a CAD design - but sadly, I'm not a mathematician. In other words, this question might sound like homework, but it's not, I promise. I have an existing circle, ...
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1answer
44 views

In a cyclic pentagon $ABCDE$, prove $\frac{a}{\sin(B+E)}=\frac{d}{\sin(C+E)}$

I've found this question online without a solution, I have tried myself and I couldn't go further than what I've mentioned in the picture. Question: My attempt So I simplified the question into ...
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1answer
31 views

Mixtilinears and Symmedians

(this is from EGMO) Prove that angles ATK and LTI are equal. The hint in the book was about symmedians. I am not sure how to prove that line segment AT is the T symmedian, angle chasing did not work, ...
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44 views

Proving that two circles in the Complex Plane intersect orthogonally?

I've been stuck on this question for the better part of a day now: Let $a$ be a non-zero complex number such that $|a| \neq 1$. Let $P$ be the point $a$ in the complex plane, and let $Q$ be the point ...
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work out points that describe a section of a circle.

Given a circle with a radius of r, knowing that two points $A, B$ are on that circle. Find a way to describe the arc between $A$ and $B$. if following information is given: the arc does not span over ...
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1answer
57 views

A circle rolling along the base of an isosceles triangle has constant arc length cut out by the lateral sides.

The problem is from Kiselev's Geometry Exercise 582: A circle of the radius congruent to the altitude of a given isosceles triangle is rolling along the base. Show that the arc length cut out on the ...
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2answers
69 views

Prove Angles are Equal [closed]

Vertex $A$ of an acute triangle $ABC$ is connected by a segment with center $O$ of the circumscribed circle from vertex $A$ height $AH$ is drawn. Prove that $\angle BAH = \angle OAC$. $ $ So I don't ...
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1answer
63 views

Find the angle θ (all the circles are tangent)

In the following figure ABCD is a side square $\alpha$, the points $P_0, P_1, P_2, P_3, Q_0, Q_1, Q_2, Q_3, X \ and \ Y$ are points of tangency, $BC \ and \ ZB$ are the diameters, respectively, of the ...

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