Circle is the locus of points equidistant from a given point, the center of the circle. The common distance from the center of the circle to its points is called radius. Thus a circle is completely defined by its center (O) and radius (R):
C(O, R) = O(R) = {x: dist(O, x) = R}.
A circle is a plane figure contained by one line, which is called circumference, and is such, that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.
What is so different between our definition and Euclid's? Strange as it may seem, Euclid did not have a notion of metric or a distance function. Nowhere in the Elements he refers to a distance between two points, but only to the length of the line segment joining the two. The whole of geometry has been developed without the idea of a 2d-distance! And so it does not seem to be necessary. The definition we gave which is the most common nowadays is the outgrowth of the culture permeated by analytic geometry. Ironically, in the modern terminology being Euclidean exactly means being in possession of a suitable metric. Euclid might have objected.
Now, returning to the 21st century, our definition is meaningful in any metric space. For some metrics the circles may look rather square. However, a circle looks like a circle in Euclidean geometry. Although the Euclidean circle has much to recommend it for, long, long ago Piet Hein has discovered that circles in the metric
dist((x1, y1), (x2, y2)) = ((x1 - x2)5/2 + (y1 - y2)5/2)2/5
are more pleasant to the eye than the Euclidean ones.
For square circles, it is relatively simple to define their length as the sum of lengths of the constituent sides. The length of the Euclidean (ordinary) circle may be approximately measured to any precision with a variety of means. A rigorous definition, nonetheless, requires a theory of limits - the fundamental notion in Calculus. The length of a circle is usually called its circumference.
In any affine space endowed with a metric, all circles are similar. Which does not mean there is similarity between circles in various metric spaces. But which does mean that, however defined, the ratio of the circumference to the radius of circles in a given metric space is constant. It may vary between spaces, though. In Euclidean space, the ratio equals 2p, named so by L. Euler.
The most common tool for constructing (Euclidean) circles is the venerable compass. But there are of course other ways. The inversion can be used to convert circular motion into rectilinear and vice versa. A circle is also traced by the point common to two lighthouse beams rotating at the same angular speed. The midpoint of a ladder sliding down a wall also traces a circle. Circle is also the locus of points having the same ratio of distances to a pair of points. Circle (a part of it actually) is also the locus of points from which a given line segment is seen under a given angle.
The line joining the center of the circle to one of its points is often called the radius-vector of that point, or just one of the radii. Likewise, the word diameter is used in two different, but close meanings. On one hand, diameter is the largest distance between any two points on a circle. As such, it equals twice the radius of the circle. For a given circle, there is exactly one and only one diameter. On the other hand, the word "diameter" designates a line segment joining a point on the circle with its opposite, i.e. the point farthest away from the point at hand. For every point on a circle there is exactly one diameter that contains this point. The length of a diameter, as a line segment, is exactly the diameter, as a number. Any diameter passes through the center of the circle and is divided by the center into two equal halves, each the length of the radius.
Two secants that meet on a circle define an inscribed angle. Two radii define a central angle.
Two intersecting chord are divided by the point of intersection into two parts each such that the products of their lengths are equal. This is known as the Intersecting Chords Theorem. The latter carries on to the intersecting secants and gives rise to the important notion of the power of a point with respect to a circle.
Triangles always have one inscribed and one circumscribed circle. Three more circles (excircles) touch the triangle's sidelines. Many other special circles are associated with a triangle: 9-Point circle, Taylor circle, Tucker circles, Adams' circle, to name a few. Archimedes has discovered his Twin Circles in the wonderful arbelos.
Some quadrilaterals have inscribed circles, and these are called inscriptible. Other quadrilaterals have concyclic vertices, and these are called circumscriptible, or cyclic. Those quadrilaterals that have both inscribed and circumscribed circles are said to be bicentric.
Surprisingly, in many respects circles behave like straight lines. Nowehere it is more apparent than in inversive geometry.
References
- T. L. Heath, Euclid. The Thirteen Books of the Elements, vol 1, Dover, 1956
- Robert Simson, The Elements of Euclid, Elibron Classics, 2005
