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Concept

Morley theorem

Morley's trisector theorem: the adjacent trisectors of any triangle's angles meet in three points that form an equilateral triangle. Discovered in 1899, it holds for every triangle whatever, and its proofs run backwards from the equilateral triangle rather than forwards from the given one.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

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 Morley's construction on a sphere. A spherical triangle with angles 76.1, 97.3, 48.8 degrees, its angle trisectors as great-circle arcs, and the inner triangle they form, with sides 0.23902, 0.23881, 0.23902.

The trisectors on a sphere almost agree

Draw Morley's construction on a sphere and the triangle the trisectors cut out is not equilateral — but its sides agree to within a tenth of a percent even when the angles add to 222°. Every step of the rotation proof survives on the sphere except one, and that one is the angle sum.

geometry · Morley How far from equilateral a curved Morley triangle is, shape by shape. First-order coefficient of Morley-triangle spread per unit angle excess on the sphere, over 820 shapes; maximum 0.00303 at angles 23, 19, 139 degrees; zero at the equilateral shape.

The price of curvature in Morley's triangle

On a sphere, Morley's trisectors meet in a triangle that is not quite equilateral, and its failure grows with the curvature the triangle encloses. For a small triangle the failure is proportional to the angle excess, with a coefficient set by the triangle's shape alone. Computed over every shape, the coefficient is nought for the equilateral triangle, at most 0.00303 for any shape with no angle below 11°, largest for obtuse triangles near 139°, 23° and 19° — and exactly the same on the hyperbolic plane, where the excess is negative.

geometry · Morley

Named alongside it

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

Angle trisectionEquilateral triangleHyperbolic geometrySpherical geometryTrigonometryAngle excessAngle sumCircumradiusCurvatureExhaustive searchLaw of sinesModular arithmetic

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