An interactive column using Java applets
Cut The Knot!
by Alex Bogomolny
Doodling and Miracles
Almost 100 years ago, Frank Morley proved a curious theorem from elementary geometry that unbelievably remained unknown until 1899. With time, the theorem became known in mathematical folklore as Morley's Miracle (Morley's Trisector Theorem is a more mundane term.)
The theorem states that
The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle.
Frank V. Morley, the youngest of Morley's sons, recollects [Oakley and Baker]
I was a school-boy when my father, who was almost forty years older than I was, sketched for me, free-hand, a pencilled diagram of the simplest form of the above-discussed theorem in plane geometry.
I tested it once with my own drawing instruments. No matter what shape of the original triangle I started with, there in its midriff was an equilateral triangle, picked out by the trisectors. It was wizard, it was weird - and it was True!
(Personally, drawing by hand, I have never succeeded to get anything that remotely resembled a polygon, let alone an equilateral triangle. If you like me, the applet below will save you from frustration you may experience watching irregular shapes emerge on paper all your efforts notwithstanding - the vertices of the triangle are draggable.)
Remarkably, Morley's 1929 paper where he mentioned the theorem explicitly for the first time, contains not a single diagram. In the middle of the paper, at the end of one of the sections, there appears the following paragraph
If we apply the theory of this section to a triangle abc, we obtain as the locus of centers of inscribed cardioids three sets of three parallel lines, forming equilateral triangles. The vertices of the triangles are the centers of the cardioids which touch a side (say bc) of the given triangle twice. If xo be such a center, then the angle xobc is a third of the angle abc. For xob is an axis of the 3 lines ab and bc twice. Thus if we take the interior trisectors of the angles of a triangle, the points where those adjacent to a side meet form an equilateral triangle.
Author David Wells wrote:
Frank Morley was studying cardioids in 1899 when he came across an extraordinary theorem, which anyone doodling with pencil and paper might have previously spotted.
After many failed attempts at doodling I feel justified to question this statement. How easy is it to spot a theorem like this? Even if instead of doodling one uses computers, is it easy? Let's run an experiment. Who knows? If we are lucky, we may be able to discover perhaps a lesser miracle. What might it be? A generalization of Morley's theorem would be an appropriate contribution to the coming centennial anniversary of the discovery of this wonderful result. J. Littlewood mentions in his Miscellany that
Erasmus Darwin held that every so often you should try a damnfool experiment. He played the trombone to his tulips. This particular result was in fact negative.
One foolish thing I could think of that was related to Morley' theorem was to replace the angle trisectors with a more general sort of lines. Divide an angle into n equal parts with (n-1) lines, remove all the lines but the extreme two - the ones which are next to the sides of a given triangle. As with Morley's Miracle, we get a triangle at the intersection of those lines. It would be nice if it were equilateral. Check your perception. Does not the triangle look equilateral to you?
Well, for small values of n (say 4 or 5) you might have been fooled. For larger values, the triangle sheds perceptibly its regular shape. If you check the "Show angles" button in the version of the applet below, you'll see that even for small values of n the conjecture looks highly improbable. So this generalization appears wrong - another negative result. (Was it not foolish enough?) However, note that (experimentally) angles in "Morley's" triangles are distributed more or less evenly with the smallest angle never below 35°. For smaller values of n, the borderline value is even higher: for
For "more or less normal" triangles, the angles differ only slightly. I suspect that doodling might have led one to think that, for
But, perhaps, not everything has been lost. Did you try checking (say out of curiosity) the small radio buttons at the right bottom portion of the applet? If you did, you might have noticed three families of concurrent lines, i.e. the lines that meet at a single point:
- PU, QV, RW
- AP, BQ, CR
- AU, BV, CW
Toying with the applet provides a convincing demonstration that in all three families the lines are indeed concurrent regardless of the value of n. The three points are in general distinct.
There are many families of concurrent lines in a triangle. The best known are the angle bisectors, medians, altitudes, and perpendicular bisectors. There are more. On the Web, Clark Kimberling of University of Evansville has collected a respectable list of such points and the corresponding families of lines. It's there that I also learned, albeit somewhat late, that "Morley's" triangles with 1/n replaced by a real r have been known for a while as the Hofstadter triangles.
First, let's see why the lines AP, BQ, CR are always concurrent. The lines (that contain the segments) AQ and AR are isogonal which simply means that they are reflections of each other in the bisector of ∠A. A similar statement holds for the pairs BP, BR and CP, CQ. Isogonal lines are featured in the construction of the Fermat and Napoleon points. It turns out [Gale] that there is a very general statement concerning isogonal lines:
Given ΔABC, and three pairs of lines a, a', b, b', and c, c' isogonal at vertices A, B, and, C, respectively. Denote
This theorem tells us that lines AP, BQ, CR are indeed concurrent. More than that, lines AU, BV, CW are also concurrent and for the very same reason, since their construction starts with the same three pairs of isogonal lines as the construction of lines AP, BQ, CR!
D. Gale notes that "the result is so simple and natural in its statement that one suspects it must have been noted long ago, but the historical trail seems to be murky." He proves a more general statement of an undoubtedly recent vintage. Surprisingly (because angle measurements are involved) the above theorem about isogonal lines falls into the framework of Projective Geometry. In that framework, concurrency of the lines PU, QV, RW is deduced from a result in Analytic Geometry illustrated by the following applet.
Let a, b, c be real non-zero numbers. Consider six lines,
The fact is that the segments connecting these pairs of lines are concurrent.
Back to Morley's triangle. That this triangle is always equilateral is utterly startling. I never heard another opinion on this account. Opinions, however, diverge on another point. Some consider the theorem beautiful but others disagree. For example, G.-C. Rota says that "... one can find instances of surprising results which no one has ever thought of classifying as beautiful. Morley's theorem, stating that ... is unquestionably surprising, but neither the statement nor any of the proofs are beautiful despite the repeated attempts to provide streamlined proofs." Oakley and Baker, on the other hand, are of opinion that "It is one of the most astonishing and totally unexpected theorems in mathematics and, jewel that it is, for sheer beauty it has few rivals." (Note in passing that the sheer existence of C. O. Oakley and J. C. Baker unquestionably refutes the first part of Rota's argument with regard to Morley's theorem.)
Immanuel Kant remarks
... the lively sensation of the beautiful proclaims itself through shining cheerfulness in the eyes, through smiling features, and often through audible mirth.
Check yourself: does the statement of Morley's theorem elicits a smile on your face? The theorem surely evoked considerable interest in the mathematical community as witnessed by a steady stream of publications - 150 by 1978 [Oakley and Baker] and more afterwards. Some proofs are direct and some "backward" that start with the equilateral triangle [Coxeter, Coxeter and Greitzer]. The latter are in general simpler. Of the latest crop, the proofs by D. J. Newman and J. H. Conway are probably the simplest. However, shortness of both proofs implicitly depends on the knowledge about Morley's configuration that, in other proofs, is extracted explicitly. Without this information (angle magnitudes in the triangles in Morley's configuration) the proofs appear as pure magic. To a student, they leave little to learn.
One of the participants in the geometry.puzzles newsgroup posted a message concerning Conway's proof: "I remember downloading a proof given by John Conway. I found it and am enclosing it below. I started going through it, but haven't finished reading it to see if I'm convinced. Any comments on this proof?" This is about a proof that takes all of 30 lines most of which are either short or sparse! Compare this to D.J.Newman: "When I read, or rather tried to read, Morley's proof of this startling theorem, I found it absolutely impenetrable. I told myself that maybe in future years I would return and then understand it. I never succeeded in that ..."
Newman writes further
The reason that all the proofs seem to be so difficult and unmotivated is probably because Morley's theorem is really only half the story. The full picture is in Figure 1 and this tells the whole story and indeed proves itself!
Figure 1 is indeed suggestive of a possible proof. In fact, Conway's proof starts with the same diagram and so does another one. However, when I am looking at the diagram, I feel sadness rather than mirth. The mystery inherent in the wonderful surprise that is Morley's discovery is completely gone. It was different when I read Morley's own account although then my mirth was tinged with melancholy. For I too had difficulty with Morley's reasoning. So what? He never set out to prove that theorem in the first place! I can only guess what he felt when an equilateral triangle emerged in his mind's eye from among the tangents to the heart shaped curves. If only we could communicate a like sensation with simpler means.
- H. S. M. Coxeter, Introduction to Geometry, Toronto: John Wiley and Sons (1969), pp. 23-25.
- H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, MAA, 1967
- D.Gale, Triangles and Proofs, The Mathematical Intelligencer, v 18, n 1, 1996. p 31-34. (Reprinted in D. Gale, Tracking the Automatic Ant, Springer, 1998)
- I. Kant, Observations on the Feeling of the Beautiful and Sublime, U. of California Press, Berkeley, 1960
- T.D.J.Kleven, Morley's Theorem and a Converse, Amer Math Monthly, 85 (1978) 100-105.
- Littlewood's Miscellany, Béla Bollobás (ed), Cambridge University Press, 1990.
- F. Morley, Extensions of Clifford's Chain-Theorem, Amer J Math, 51 (1929) 465-472.
- D. J. Newman, in The Mathematical Intelligencer, v 18, n 1, 1996. p 31-32
- C. O. Oakley and J. C. Baker, The Morley Trisector Theorem, Amer Math Monthly, 85 (1978) 737-745.
- G.-C. Rota, Indiscrete Thoughts, Birkhäuser, 1997
- D. O. Shklyarsky, N. N. Chentsov, Y. M. Yaglom, Selected Problems and Theorems of Elementary Mathematics, v. 2, problem 97, Moscow, 1952.
- D. Wells, The Penguin Dictionary of Curious And Interesting Geometry, Penguin Books, 1991
Copyright © 1996-2018 Alexander Bogomolny