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Morley — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Trisect every angle, and an equilateral triangle appears. A triangle with angles 78°, 54°, 48°, its six angle trisectors, and the triangle whose corners are where the trisectors nearest each side meet. That inner triangle is equilateral, which is Morley's theorem.

    Three trisectors and a triangle nobody expected

    Cut every angle of a triangle into three. The trisectors nearest each side meet in three points, and those three points are always the corners of an equilateral triangle — for every triangle there is, with no exceptions and no reason anybody finds obvious.

    part 1 · geometry
  2. 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.

    part 2 · geometry
  3. 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.

    part 3 · geometry
  4. 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.

    part 4 · geometry

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