In the picture above, whatever the shape of the outside triangle, the purple triangle is always equilateral – if the two inside lines at each corner of the outside triangle trisect the angle at that corner (i.e. divide it into three equal bits).
This result is called “Morley’s Miracle”.
Lola Behanzin, Taija Williams, and Serene Williams proved it, given nine “thin” triangles which have angles (α, β+60, γ+60), (β, γ+60, α+60), (γ, α+60, β+60); which also have their short sides (the sides connecting the “+60” angles) all equal; and for which 3α+3β+3γ=180.
You can fit together the nine “thin” triangles around a “small” equilateral triangle, with overlapping but neatly so they overlap exactly to form a “big” triangle with angles 3α, 3β, 3γ. The “small” equilateral triangle is then the triangle of the meeting points of the trisector lines of “big” triangle. This can be done for any α, β, γ.
So any shape of big triangle can be built up round a small equilateral triangle made by the meeting point of the big triangle’s trisectors. Conversely, every triangle’s trisector lines meet to form an equilateral triangle.
But do the “thin” triangles really fit together so neatly, to form one large triangle? Or does it just look like that?
What we have to prove is that the overlap area (marked dark blue) of the (α, β+60, γ+60) (marked green) and (β, γ+60, α+60) (marked light blue) triangles in the picture below really is a triangle, not a kite. To do that all we need to prove is that the angles marked in it add up to 180.
If the thin triangles fit together neatly along that edge to form a straight line from the α-angles corner to the β-angles corner, then, by symmetry, the thin triangles are going to fit together neatly all round to form a big triangle as required.
But the angle marked x in red is given by the excess of the total of all the angles round the point in the middle of this diagram over 360.
x = α + 60 + α + 60 + 60 + β + 60 + β + 60 – 360
= 2α + 2β – 60
So the angles marked in the dark blue area add up to
2α + 2β – 60 + γ + 60 + γ + 60 = 2(α+β+γ) + 60 = 180 ◾
A video presentation:
The “Morley Miracle” result was first discovered at the end of the 19th century, by Frank Morley. Until 1995 all known proofs were long and complicated. The idea for the proof which Lola, Taija, and Serene rediscovered (with a bit of help provided by having the triangles cut out for them) was discovered by John Conway.