# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...
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 ...