# chord

Chord of an arc.

A chord is a straight line that joins two points on a curve. The chord of an arc is a line joining its two extremities. Most commonly, chord is used to mean
a straight line segment joining, and included between, two points on a circle, ellipse, parabola,
etc. In the restricted sense of a chord of circle the term first appears
in English in 1551 in Robert Recorde's *The
Pathway to Knowledge*: "Defin., If the line goe crosse the circle,
and passe beside the centre, then is it called a corde, or a stryngline."

If we draw a series of parallel chords in any conic, the line through their middle points is called a diameter, and a line parallel to the chords which passes through the extremity of the diameter is a tangent to the curve. Hence in the circle (1) a diameter is perpendicular to the chords which it bisects, and also (2) to the tangent at its extremity; (3) the three chords of intersection of any three circles meet in a point. In any conic section the tangents at the ends of any chord meet in the diameter which bisects the chord.

Some surprising results emerge from moving chords. For example, take a chord
in a circle *C*, and slide the chord around the circle so that the
midpoint of the chord traces out a smaller concentric circle. Call the area
between the two circles *A*(*C*). Now do the same thing with
a larger circle *C'* but with the same length chord. Is *A*(*C'*)
larger or smaller than *A*(*C*)? Surprisingly, they are the
same. In other words *A*(*C*) doesn't depend on what circle
we start with, only the length of the chord. An even more amazing fact is
that if we slide a chord of fixed length around any convex shape *C* so that the chord's midpoint traces out another figure *D*, the area
between *C* and *D* doesn't depend on what shape we started
with.