Concept

Law of sines

The identity that in any triangle each side divided by the sine of the angle opposite it gives the same value, twice the circumradius. It converts a statement about angles into one about lengths, and is the usual way a trigonometric proof of a geometric theorem gets started.

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

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

Conway's seven pieces, pulled apart and measured. A triangle with angles 78, 54, 48 degrees divided along its angle trisectors into an equilateral centre, three pieces touching it along its sides and three along the outer sides, spread apart, with each corner's angle labelled as a third of an outer angle plus a multiple of sixty degrees.

Seven pieces and an equilateral middle

Morley's theorem has two proofs worth knowing, and they run in opposite directions. The trigonometric one starts from the triangle and computes each side of the inner one as 8R sin α sin β sin γ, symmetric in the three angles. Conway's starts from an equilateral triangle, builds six pieces round it from their angles alone, and shows they fit — so the triangle they make is whatever triangle was wanted, and its middle is equilateral because it was built that way.

geometry · Morley

Named alongside it

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

Angle trisection Chord Circumcircle Circumradius Equilateral triangle Inscribed angle Morley theorem Ptolemys theorem Rational distance Sine Trigonometry

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