A

circleis a plane figure contained by a single line [which is called acircumference], (such that) all of the straight linesradiatingtowards [the circumference] from one point amongst those lying inside the figure are congruent to one another.

Let \(A,B\) be two different points in a plane \(\mathcal P\), and let \(d(A,B)\) denote their Euclidean distance (i.e. the length of the segment \(\overline{AB}\)). A **circle** is a plane figure consisting of points of the plane \(D\), which have a smaller or equal distance from \(A\), formally

\[\text{Circle}:=\{D\in\mathcal P:~d(A,D)\le d(A,B)\}.\]

Any segment \(\overline{A,D}\) with maximum possible distance \(d(A,D)=d(A,B)\) is called the **radius** of the circle.

The **circumference** of the circle is the boundary of the circle, i.e. all points \(D\) in the plane, which have the maximum possible distance:

\[\text{Circlumference}:=\{D\in\mathcal P:~d(A,D) = d(A,B)\}.\]

A circle with a radius \(\overline{AB}\). In the figure, one of the infinitely many points \(D\) with \(d(A,D)\le d(A,B)\) is marked.

| | | | | created: 2014-06-14 19:46:05 | modified: 2019-08-20 20:41:32 | by: *bookofproofs* | references: [626], [628], [6419]

[626] **Callahan, Daniel**: “Euclid’s ‘Elements’ Redux”, http://starrhorse.com/euclid/, 2014

[6419] **Fitzpatrick, Richard**: “Euclid’s Elements of Geometry”, http://farside.ph.utexas.edu/Books/Euclid/Euclid.html, 2007

[628] **Casey, John**: “The First Six Books of the Elements of Euclid”, http://www.gutenberg.org/ebooks/21076, 2007

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