Chord
A chord is a line segment with both endpoints on a circle, with the diameter being the longest possible chord.
Formula
\text{chord length} = 2r\sin(\theta/2) \text{ (where } \theta \text{ is the central angle)}
Definition
A chord is a straight line connecting two points on a circle; unlike a diameter, it does not have to pass through the center, though the diameter is always the longest possible chord. Its length for a central angle $\theta$ in a circle of radius $r$ is $2r\sin(\theta/2)$, and the perpendicular from the center to a chord always bisects both the chord and the arc it subtends. The power of a point $P$ with respect to a circle (center $O$, radius $r$) is $d^2-r^2$ where $d=|OP|$; the intersecting chords theorem states that if chords $AB$ and $CD$ meet at $P$ inside a circle, $AP\times PB=CP\times PD$, an instance of this same power.
Example
A line drawn across a circle that does not pass through the center is a chord; the diameter is a special chord that cuts the circle in half, and a slice cut off by a chord is called a segment. For radius $10$ and central angle $60^\circ$: chord $=2\cdot10\sin(30^\circ)=10$, since the $60^\circ$ central angle case forms an equilateral triangle. For a point $P$ inside a circle with two chords through it, $AP=3$, $PB=4$, $CP=2$: then $PD=AP\times PB/CP=12/2=6$, and the power of $P$ is $-(AP\times PB)=-12$, negative since $P$ is inside.
Key Insight
Imagine a bowstring stretched across a curved bow, that is the image behind the word "chord" (also used in music for notes played together). The power of a point is an invariant, the same product for any chord or secant through that point, depending only on the circle and the point, and this invariance underlies radical axes (the locus of equal power with respect to two circles), used in classical construction problems.