In the picture above, 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 with angles (α, β+60, γ+60), (β, γ+60, α+60), (γ, α+60, β+60); their short sides (the sides connecting the “+60” angles) all equal; and 3α+3β+3γ=180.
Here is how Lola, Taija, and Serene showed that the nine “thin” triangles fit together (with some overlapping) to form a “big” triangle with angles α, β, γ and, inside it, a “small” equilateral triangle made by the meeting points of the trisector lines of the angles α, β, γ. This can be done for any α, β, γ. So any shape of triangle can be built up round an equilateral triangle made by the meeting point of the trisectors. So every triangle’s trisector lines meet to form an equilateral triangle.
But maybe the triangle just looks roughly equilateral? Here is how Lola proved that, if you can position the “thin” triangles exactly, all the angles of the “little” triangle inside must be 60 degrees exactly.
Here’s another explanation of the proof.
The proof works by approaching the problem from an imaginative and unexpected viewpoint.
Instead of starting with the outside triangle and working inwards, he starts from an equilateral triangle and proves that, for any three angles 3α, 3β, and 3γ adding up to 180 degrees, you can construct a “trisector triangle” with those angles around that equilateral triangle.
Such imaginative ways of looking at problems are at the core of much of the best mathematical thinking. The proof is an excellent example to learn from.
How could anyone have thought it up? If you start from an equilateral triangle in the centre, you can see from the argument below that the “thin” triangles (with angles α, β, and γ at the vertices of the “outside” triangle, and connected from those vertices to the “inside” triangle by trisector lines) must have angles (α, β+60, γ+60), etc. Then you turn the argument round to show that for any 3α, 3β, 3γ you can construct “thin” triangles around the “inside” equilateral triangle with the angles (α, β+60, γ+60), etc.
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.