Three trisectors and a triangle nobody expected

Take any triangle. Cut each of its three angles into three equal parts, giving six rays. Two of those rays are adjacent to each side of the triangle; take the pair adjacent to each side and find where they meet. Three sides, three meeting points.

The inner triangle is equilateral. The outer triangle has nothing equal about it, no symmetry, and no relationship between its angles, and the inner one is equilateral anyway.

Why this is startling #

Three features of the statement are each individually odd, and together they explain why the result was not found until 1899 — long after every other elementary fact about triangles had been catalogued.

Trisection is not a classical operation. Angle bisectors are constructible with straightedge and compass and appear in Euclid; angle trisectors are not constructible and cannot be drawn by the classical tools at all. A theorem about them could not have been discovered by anybody drawing figures in the classical way, because the figure cannot be drawn.

The nearest trisectors are an arbitrary-looking choice. Each vertex has two trisectors; the theorem uses the one nearer each side, and taking the other one instead gives a triangle that is not equilateral. Nothing about the statement suggests which choice would work.

The conclusion has no visible relationship to the hypothesis. The outer triangle’s three angles are three independent numbers; the conclusion is that three lengths, computed from them by a construction involving four intersections, are equal. There is no reason for an identity to hold, and the identity does hold for every triangle.

There is a fourth oddity, of a different kind. The theorem is named for Frank Morley, who found it around 1899 while studying the curves tangent to the trisectors of a triangle — a much larger investigation in which the equilateral triangle is one visible corner. He did not publish it as a theorem about triangles; it circulated by word of mouth among his friends for two decades and was rediscovered several times before appearing in print. A result that any competent geometer of any century could have verified in an afternoon sat unrecorded because nobody had reason to try the construction.

The sweep #

A theorem that holds for every triangle can be tested on a great many of them, and the space of triangle shapes is small enough to cover.

A triangle’s shape is settled by two of its angles, since the third is what is left of a half turn. So the space of shapes is itself a triangle, and covering it means walking a grid over that triangle. The figure takes every shape on a grid with no angle below 12°, builds the trisector triangle, and measures the ratio of its longest side to its shortest.

Every one comes out at 1, to within a part in a hundred trillion. That is not a proof and it is a great deal of evidence: 153 triangles of every shape, and the deviation is at the level where double-precision arithmetic stops being able to tell the difference.

Reading the shape space as a triangle is worth a moment, since it is the same device used to sweep any two-parameter family. Each dot’s two coordinates are two of the angles, the third is whatever is left of 180°, and the region of valid shapes is bounded by the three conditions that each angle be positive — so the space of triangles is itself a triangle, and a grid over it covers every shape exactly once. The equilateral case sits at the centre and the degenerate ones on the boundary, which is the region the sweep stays clear of. Sweeping a parameter space rather than a list of examples is what turns a set of instances into something worth calling a search, and it is the same move as the sweep over every triple of lattice points one field over.

The control #

A sweep that reports success on everything is worthless unless it is capable of reporting failure, and the way to establish that is to feed it something false.

The same code, on the same triangles, is run on the other trisector at each vertex — the one further from the side rather than nearer. That construction is as natural-looking as the theorem’s, it produces three intersection points in exactly the same way, and its inner triangle is not equilateral: over the same sweep, the longest side reaches 1.40 times the shortest.

So the measurement can tell an equilateral triangle from a nearly-equilateral one, and the fact that it reports equality for the real construction means something.

The obvious control is bisectors rather than trisectors, and it is useless — the three angle bisectors of a triangle all meet at one point, so the three intersection points coincide and the inner triangle has three sides of length zero. Zero, zero and zero are equal, and the test passes for a construction that produced nothing at all. That is worth recording: the natural control fails to be a control, because a degenerate answer satisfies an equality test.

The general lesson is one about how a check should be chosen. A control has to be a case the machinery is supposed to reject, which means it must be false and must not be false in a way that accidentally satisfies the test. The bisector construction is false in exactly the wrong way — it collapses — and the far-trisector construction is false in the right way, producing three genuine points at genuinely different distances. Choosing the second over the first is the kind of decision that separates a check from the appearance of one, and it is the same discipline the site’s proof gate applies when it feeds every assertion something wrong and requires a complaint.

The construction the tools cannot perform #

The trisectors are worth a section of their own, because their unconstructibility is the reason the theorem is late and is also a good example of what an impossibility proof looks like.

What two points can build is settled by algebra rather than by ingenuity. Each new point is the crossing of lines and circles whose equations have coefficients in the field built so far, so its coordinates satisfy an equation of degree at most two over that field. Every constructible number therefore sits at the top of a tower of quadratic extensions, and its degree over the rationals is a power of two.

Trisecting a 60° angle would require constructing cos 20°, which satisfies an irreducible cubic. Three does not divide any power of two, so the number is not constructible and the trisection is impossible — not difficult, impossible, and for a reason with no geometry in it.

That is why Morley’s figure could not have been drawn in 1830 and can be drawn now. The rays in these pictures are placed from computed angles, which is a tool Euclid did not have and which changes nothing about the theorem.

What a proof looks like #

There is no proof in a picture here, and it is worth saying what the proofs are like.

The direct approach is trigonometry. Label the outer angles 3α, 3β and 3γ, so α + β + γ = 60°. The law of sines in one of the small triangles gives one side of the inner triangle as a product of sines, and after enough manipulation each side comes out as

$$8R\sin\alpha\sin\beta\sin\gamma,$$

where R is the circumradius of the outer triangle. The expression is symmetric in α, β and γ, so all three sides are equal, and the theorem falls out. It is a correct proof and it explains nothing: the symmetry appears at the end of a page of manipulation and is not visible at the start.

The proof that explains is Conway’s, and it runs backwards. Start with an equilateral triangle and build the outer triangle around it from six triangles whose angles are chosen to fit, then check that the assembled figure has the required angles at the outer corners. Every piece fits because the angles were chosen to make them fit, and the construction produces a triangle with any prescribed angles. Since a triangle is determined up to similarity by its angles, the assembled figure is the one started with, and its inner triangle is equilateral because it was built that way.

That is a genuinely different kind of argument — build the answer and show it can be built for any input — and it is the standard remedy when a direct proof consists of an identity nobody can see the reason for.

What it belongs with #

Triangle geometry produces coincidences of this kind in quantity, and it is worth putting Morley’s beside two on this site.

The nine-point circle and the Euler line are the other two standard surprises, and both are constructible with classical tools, which is why both were found earlier — the nine-point circle in the 1820s, the Euler line in 1765. Morley’s needed a construction Euclid’s tools cannot perform, and it arrived a century after the others.

There is a common feature worth naming. In each case a construction defined by local conditions at the corners produces a global relation — collinearity, concyclicity, equality of three lengths — and in each case the relation is exact rather than approximate. Whether there is a single reason behind the family is a question the subject has never quite answered; the modern view is that many of them are shadows of statements about cubic curves, and that view explains some and not others.

The reason these keep being found is worth a sentence too. A triangle has three degrees of freedom up to similarity — really two, since scale does not matter — and any construction on it produces objects depending on those two numbers. A claimed coincidence is then an identity between functions of two variables, which is a strong constraint and usually false; when one turns out to be true, it holds for every triangle at once, and there is nothing in between. That all-or-nothing character is why this corner of geometry consists of a list of exact statements rather than of approximations with error terms, and why the exhaustive sweep over the shape space is such good evidence: a claim of this kind that survives a hundred and fifty shapes is either true or false for a reason no drawing would reveal.

An angle that does not depend on where it is #

There is one classical theorem the trisector construction leans on without saying so, and it is worth surfacing.

Every proof of Morley’s theorem, trigonometric or synthetic, moves angles around: an angle established at one vertex is transported to a triangle elsewhere in the figure and used there. The licence for that is the family of results about angles that do not depend on where they stand, together with the fact that the angles of a triangle add to a half turn.

That second fact is where the α + β + γ = 60° in the trigonometric proof comes from, and it is the only global constraint in the whole problem. Everything else is local — each vertex is trisected independently, each intersection is computed from two rays — so the one global relation is carrying the entire theorem. That is a useful thing to notice when a result looks like a coincidence: the identity that makes it work is usually the one constraint tying the local data together, and here there is only one candidate.

What it costs #

The construction is four intersections of lines, each a division of two determinants, so building one inner triangle costs a few dozen arithmetic operations. The sweep runs 153 of them twice — once for the theorem and once for the control — which is nothing, and it is why the check can be redone every time the picture is drawn.

The interesting cost is in the drawing rather than the arithmetic. The trisectors cannot be constructed with compass and straightedge, so a figure showing them has to compute the angles numerically and draw the rays from those numbers. Every drawing of Morley’s theorem in every book is in that position: it is a picture of something the classical tools cannot produce, drawn by a tool that can.

That is a mild irony worth stating. The theorem is elementary — the statement uses nothing beyond angles and intersections — and it belongs to a class of figures Euclidean geometry cannot draw. Its elementary character and its unconstructibility are not in tension, because constructibility is a fact about a particular set of tools and not about the difficulty of a statement.

What the picture cannot show #

The sweep covers triangles with no angle below 12°, and the exclusion is not cosmetic. As a triangle degenerates toward a straight line the inner triangle shrinks and the intersection points become numerically ill-conditioned — the lines being intersected are nearly parallel, so their crossing point moves a long way for a small change in the angles. The theorem holds there too; the arithmetic simply stops being able to demonstrate it.

Nor can any figure show the theorem. Every dot in the sweep is one triangle, and there are infinitely many. What the sweep establishes is that no counterexample exists at the resolution examined, which is a different and much weaker claim than the theorem’s.

And the inner triangle’s orientation is not shown to be anything in particular. It is equilateral, and there is more to say about it — its sides are parallel to particular directions determined by the outer triangle, and it has a known relationship to the circumcircle — none of which is visible in a picture that only compares three lengths.

The ladder from here #

Rungs above: the trigonometric proof written out, with the sine product that makes the symmetry appear. Conway’s construction, which builds the outer triangle from the inner one. The eighteen Morley triangles that arise from taking the other trisectors and their extensions, of which the equilateral ones are a subfamily. The orientation of the inner triangle, and the direction its sides point. Morley’s original result, which was about a much larger family of curves tangent to the trisectors and of which this is the visible corner. The trisection impossibility itself, and why an angle cannot be divided in three with the classical tools. And the analogous statements in non-Euclidean geometry, where the theorem is false.

The shape of the idea #

The essay is really about what evidence a picture can supply, and Morley’s theorem is a good test case because it is exactly the kind of statement a picture cannot prove and can strongly support.

Three things were done here. The construction was carried out exactly rather than eyeballed, so the three lengths are computed from the coordinates the intersections produced. It was carried out over the whole space of shapes rather than on one example, so the evidence is about the theorem rather than about a figure. And the same machinery was run on a construction that is false, so the measurement is known to be capable of reporting a failure.

None of that is a proof and all of it is necessary. A drawing of one triangle with three tick marks on the inner one would be a picture of a claim; a sweep with a control is a picture of a search. The habit this site runs on is that a figure should be able to fail, and the clearest way to demonstrate that it can is to show it failing on purpose.