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.
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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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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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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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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 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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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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Modifying angle at the centre is double the angle at the circumference so that it has a converse.
I'm trying to find out under what condition does the converse of the theorem hold.
My attempt:
If we have $\angle{AOC}=2\angle{ABC}$, $OA=OB$, and $O$ is on the same side of line $AB$ as $C$, does ...
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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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What is the longest side of a triangle inscribed in a given circle and circumscribed about another?
Here's a problem I just came up with :
(O) and (W) are two circles with radii of 9 and 4 respectively :
The question I asked myself is :
Among all the triangles inscribed in (O) and circumscribed ...
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Does this sequence of eights converge, and if so, to what?
In the diagram, circles of the same color have the same radius. Wherever things look tangent, they are tangent. The smallest circles have radius $1$.
Do the radii converge, and if so, to what? (closed ...
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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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Finding a parameterization of the intersection between the plane $n_x x+n_y y+n_z z=0$ and the unit sphere
Find a parameterization of the intersection between the plane $n_x x+n_y y+n_z z=0$ and the unit sphere $x^2+y^2+z^2=1$.
Stuck a little on this
Set the equations equal to each other and rearrange:
$$...
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Prove concurrent lines in a right angled trapezium
Let $ABCD$ be a right-angled trapezium with $BC \parallel AD$ and $CD \perp AD, BC$. Let $\Gamma_A$ be the circle centered at $A$ with radius $AD$, and let $\Gamma_B$ be the circle centered at $B$ ...
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