Matpic
mathematics, in pictures
Concept

Angle trisection

Dividing a given angle into three equal parts. It cannot be done with straightedge and compass alone, proved in 1837, but the trisectors themselves are perfectly good lines, and some of geometry's best theorems are about where they meet.

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

Also named here as morley theorem, trigonometry — the same set of essays touches all of them, so they are one junction rather than several.

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 Twenty-seven choices of trisector, and the eighteen equilateral triangles. Twenty-seven small panels, one for each choice of trisecting line at each corner of a triangle, each drawing the triangle and the triangle the chosen lines cut out; the eighteen equilateral ones are marked.

Eighteen equilateral triangles

Every angle of a triangle has three trisectors, not one, once the angle and its outside are both counted. Choosing one at each corner gives twenty-seven ways to cut out a triangle, and eighteen of them give an equilateral one. The nine that fail are exactly the choices whose labels add to 2, 5 or 8 — and all eighteen equilateral triangles have their sides in the same three directions, fixed by a third of the difference between two angles.

geometry · Morley

Named alongside it

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

Equilateral triangleMorley theoremTrigonometryCircumradiusExhaustive searchLaw of sinesModular arithmetic

All concepts