Chord — where it appears
Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.
The curve of the average, and the average of the curve
A curve that bends upwards keeps every one of its chords above it. That single fact, applied to a weighted average instead of a midpoint, turns into an inequality that produces the arithmetic–geometric mean inequality, Cauchy–Schwarz and the entropy bound as special cases.
Every side measured by one diameter
In any triangle, divide each side by the sine of the angle opposite it: the three answers are equal. That much is the law of sines, and it is usually left there. The common answer has a name — it is the diameter of the circle through the triangle's corners — and the reason is the inscribed angle once more. It is also why the sine was first a half-chord, why Ptolemy's theorem is the addition formula, and why a circle can hold infinitely many points all at rational distances from each other.
Wings that cut a chord equally
Take a chord of a circle and its midpoint. Draw any two more chords through the midpoint, join their ends crosswise, and the two crossing lines — the butterfly's wings — cut the first chord at equal distances from the middle. The proof is the inscribed angle and the power of a point working together; move the point off the middle and what survives is a law about reciprocals; replace the circle by any conic and the theorem does not notice.
The sine of one degree
Every table of sines ever made by hand needed the sine of one degree, and no construction with ruler and compass can produce it: one degree is a third of three, and thirds are what the constructions cannot make. Ptolemy fenced it in with an inequality and got it right to the precision his table carried. Thirteen centuries later Jamshīd al-Kāshī wrote the triple-angle formula as a rule for improving a guess and let it run, gaining almost three decimal places at every turn — a cubic solved without Cardano, by doing the same thing again.
What a random chord is
Draw a chord of a circle at random. How often is it longer than the side of the inscribed equilateral triangle? Pick two random points on the rim and the answer is 1/3. Pick a radius and a random point on it, and the answer is 1/2. Pick a random midpoint in the disc, and the answer is 1/4. All three calculations are correct, and Bertrand published them in 1889 to warn that the phrase had no meaning. Eighty-four years later Jaynes argued that one answer is right after all.
Named alongside it
The objects these essays reach for when they reach for this one.
SineInscribed angleSymmetryArithmetic meanCentroidCircleCircumcircleCobwebConicConstructible numberConvexityCubic