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mathematics, in pictures
Concept

Chord — where it appears

A straight segment joining two points of a circle or other curve. On a circle its length is fixed by the angle it subtends, and the products of the pieces of chords through one point are all equal, which is the power of that point.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A chord of x², and the curve under it. The curve x² with one chord drawn across it, the region between them shaded, and the midpoint heights of both marked. The comparison is computed at four hundred sample points.

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.

analysis · Convexity Each side over the sine of the opposite angle is the diameter. A triangle inscribed in a circle, with the diameter from one corner drawn and joined to a second corner to make a right-angled triangle; the angle opposite the side at the far end of the diameter equals the triangle's own angle opposite that side.

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.

geometry · Inscribed angle The butterfly: wings that cut a chord equally. A circle with chord PQ, its midpoint M, two chords through M, and the wings AD and BC meeting PQ at X and Y, each 0.476 from M.

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.

geometry · Inscribed angle

Named alongside it

The objects these essays reach for when they reach for this one.

Inscribed angleSineArithmetic meanCentroidCircleCircumcircleConicConvexityGeometric meanInvariantJensen inequalityLaw of sines

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