Questions tagged [circles]
For elementary questions concerning circles (or disks). A circle is the locus of points in a plane that are at a fixed distance from a fixed point. Use this tag alongside [geometry], [Euclidean geometry], or something similar. Do not use this tag for more advanced topics, such as complex analysis or topology.
6,742 questions
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Three tangent circles: prove that a line tangent to two of them goes through the top of the third one
Circles $C_1$ and $C_2$ are tangent to and above a horizontal line, and externally tangent to each other.
Circle $C_3$ is above and externally tangent to $C_1$ and $C_2$.
Prove that the line tangent ...
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What is the length of side AB in quadrilateral ABCD circumscribed about semicircle (O)?
Here's a problem I just came up with :
A semicircle (O) is inscribed in a quadrilateral ABCD , as shown in the figure.
If sides AD , DC , CB measure 17 ; 16 and 14 respectively, what is the length of ...
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Four circles in a star: seeking intuitive proof
The diagram shows a regular pentagram and three inscribed circles, and a dashed line tangent to the two smaller circles.
I proved that there exists a (red) circle that is tangent to the the other ...
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Sum of parallel chords given fixed spacing between chords
We manufacture circular objects featuring evenly spaced parallel lines. Although the image below illustrates the general concept, our design does not include a chord passing through the center.
I am ...
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What is the term for a chain of arcs that share endpoints?
Adjacent arcs are defined as arcs of the same circle that do not overlap and share exactly one endpoint.
The chain of arcs used to approximate the length of a cycloid curve fails the first part of ...
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Prove that $H$ lies on the incircle.
Problem: Let $ABC$ be a scalene triangle inscribed in a circle $\omega$. Let there be an incircle with center $I$. Let $BI \cap \omega=G, CI \cap \omega=F$. $FG$ intersects the tangent at $A$ at the ...
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Show that the locus of images of $A$ under inversion from varying circles passing through $B$ and $C$ is the Apollonius circle
I got this problem from a friend:
Let $\omega$ be an arbitrary circle passing through points $B$ and $C$. Show that the image of $A$ under an inversion about $\omega$ lies on a circle.
Here's a ...
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In the family of circles tangent to two intersecting circles, each pair generates 3 straight lines : prove that they concur on the radical axis
Fig. 1 : A global view on family $\frak{F}$ of circles internaly or externaly tangent to 2 (fixed) intersecting circles
Fig. 2. Being given two intersecting circles in $A,B$ with centers $P$ and $T$, ...
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Circumcircle of triangle ABC has diam. AD. Tangent in D intersects BC in P. $M=AB\cap PO$ and $N=AC \cap PO$ verify $OM=ON$. Alternate proof?
Here is an apparently simple question, in fact rather puzzling, that has been asked some days ago ; it had been closed by lack of work. I have decided to re-publish it with a solution, and I am asking ...
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Can this metric relation be proven synthetically?
In the attached figure, circle $O$ passes through vertex $A$ of $\triangle ABC$, intersecting sides $AB, AC,$ and $BC$ at $\{A, M\}, \{A, N\},$ and $\{P, Q\}$.
Using complex numbers, I found:
$|AB| \...
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What kind of curve is made of half circles?
I have this curve. It's definitely no sine or cosine. It consists of half circles. How do you call it and how do you describe it mathematically?
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Seven circles in a rectangle: show that two of them are congruent
The diagram shows seven circles in a rectangle. Wherever things look tangent, they are tangent. (The tangencies imply that the two largest circles are congruent, and the top-left and top-right circles ...
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Trying to calculate necessary turning radius for a circular arc given two cartesian points, (start and end) and two headings (start and end)
I have a question that FEELS simple, yet I'm unable to articulate it. Probably because I'm in no way a mathematician. This is in no way math homework, but here's the problem.
Say you have an initial ...
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An unexpected equilateral triangle (from a diagram with three circles in a triangle)
In equilateral $\triangle ABC$, $D$ is on $\overline{AB}$, $E$ is on $\overline{AC}$, and $M$ is the midpoint of $\overline{BC}$.
There are three congruent circles: the incircle of $\triangle ADE$, ...
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Orthocentre and circumcentre at $(9,5)$ and $(0,0)$
Let $ABC$ be a triangle having respectively orthocentre and circumcentre at $(9,5)$ and $(0,0)$. If the equation of side $BC$ is $2x-y=10$, then find the possible co-ordinates of vertex $A$.
MY ...
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