radius, diameter, chord, secant and tangent

A circle is the set of all points at one fixed distance from a centre — the rim you draw by swinging a compass. Around and across that rim, geometers give special names to the lines and segments you can draw, just as a map names a city's streets. Learning this vocabulary is the first step to every theorem about circles.

Here is the dictionary. The radius is a segment from the centre to any point on the circle (and 'radius' also names that common distance, r). A chord is a segment whose two endpoints lie on the circle — a straight cut across it. The diameter is the longest chord, one that passes through the centre; its length is twice the radius, 2r, and it splits the circle into two equal halves. A secant is a full line that cuts the circle at two points (so a chord is the part of a secant inside the circle). A tangent is a line that touches the circle at exactly one point, called the point of tangency, grazing it without crossing. These are the raw ingredients; arcs, sectors, central and inscribed angles, and the power of a point are all built from them.

A point worth getting straight: 'radius' and 'diameter' name both a segment and a number (its length), and authors slide between the two — context tells you which. A common slip is to call any chord through 'the middle-ish' a diameter; only a chord through the exact centre is a diameter. Another is to confuse a secant (a whole line, two crossing points) with a tangent (one touching point); the difference of one versus two intersection points drives much of circle geometry, including the power-of-a-point relations and Thales' theorem.