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.
6,765 questions
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How to prove that four points are concyclic under these given conditions?
I've stumbled upon the following problem that made me curious for quite a while. Below is one of many possible diagrams of the problem in question.
Given that $ABCD$ is a parallelogram with acute ...
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How to map square roots as a linear progression on a circle?
I have been exploring geometric constructions with roots. It seems that the fact that the circle is the locus of precise square root dispositions is often overlooked, as I haven't seen it used much in ...
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Concyclicity of Fermat points in a cyclic quadrilateral
About six months ago I came up with a nice property related to Ferma points and circular quadrilaterals, but I couldn't prove it:
Let $ABCD$ be a cyclic quadrilateral. For each vertex, consider the ...
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Non-concyclicity of the circumcenters of complementary triangles in a quadrilateral
About a year ago, I discovered a property related to quatrains, but I haven't been able to prove it. The property is rather strange, and I don't know where to begin proving it. If anyone can help me, ...
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Calculating the radius of a semicircle inscribed in a triangle
I've just rediscovered an old problem I first encountered over half a century ago: It's the problem number $191$ from Julius Petersen's famous work, whose initial statement is as follows:
In a given $...
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Coaxality of three circles in a complete cyclic quadrilateral
Let $ABCD$ be a cyclic quadrilateral. Consider the three intersection points of its pairs of opposite sides:
$P = AB \cap CD$,
$Q = AC \cap BD$,
$R = AD \cap BC$.
These three points form the diagonal ...
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How do we know which root of the decic is the radius of the circle?
I. Three circles
In this post, the OP asked for the radius $r$ of the central circle,
if the radii of the other three are $(d,e,f)=(3,6,7)$, respectively. Heropup gave the answer $r \approx 4.9648$ ...
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Prove that $PM, NH, QI$ are concurrent
Given an ellipse $E: \frac{x^{2}}{4} + y^{2} = 1$ with left focus $F$.
Let $P$ be a point on the circle $O: x^{2} + y^{2} = 4$.
From $P$, draw two tangents $PM$ and $PN$ to the ellipse $E$, with ...
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Three Tangents Lemma Using Directed Angles
The following is Lemma $1.44$ in Euclidean Geometry in Mathematical Olympiads by Evan Chen.
Let $ABC$ be an acute triangle. Let $BE$ and $CF$ be altitudes of
$\triangle ABC$, and denote by $M$ the ...
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A Surprising Radii Equality in a Circle Chain Tangent to a Parabola
A previous problem related to this one is here .
As shown in the figure, the green curve is a parabola. Three circles, each tangent to its neighbors, are all tangent to the parabola. The small red ...
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Circles Tangent to a Parabola and Two Adjacent Circles
As shown in the figure, the green curve is a parabola. Take $y^2=4x$ as an example. The radius of the first black circle is $2$, and the second black circle is tangent to both the first one and the ...
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What is the measure of angle $A$, given $|AB|$ and $|AC|$ , such that the incircle of $\triangle ABC$ is maximal?
Here's a problem I just came up with:
I considered a triangle $\triangle ABC$ such $|AB|=11$ and $|AC|=5$.
I asked myself the following question:
What must angle A be when the incircle of ABC has a ...
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Prove that $\angle POB = 2\angle PBO$
The point $A$ is on circle with centre $O$. $OA$ is extended to $C$ s.t. $OA=AC$, and $B$ is the midpoint of $AC$. The point $Q$ is on the circle such that $\angle AOQ$ is obtuse. The line QO meets ...
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Find the minimum value of $\frac{BX}{FC}$ and find the triangles $\triangle ABC$ when it is possible
In triangle $\triangle ABC$, the points $L, M$ are the midpoints of $BC$ and $CA$, respectively, and $CF$ is the altitude from $C.$ The circle through $A$ and $M$ which $AL$ is tangent to at $A$ meets ...
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Transcendental equation $n\sin\left(\frac{2\pi}{n}\right) = 2$ — is the solution a known constant? [closed]
While exploring the area of a regular $n$-gon inscribed in a unit circle ($r=1$), I arrived at the equation:
$$\frac{n}{2}\sin\left(\frac{2\pi}{n}\right) = 1$$
or equivalently:
$$n\sin\left(\frac{2\pi}...
