Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
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
Frank V. Morley, the youngest of Morley sons, recollects [Oakley and Baker]
(Personally, drawing by hand, I have never succeeded to get anything that remotely resembled a polygon, let alone an equilateral triangle. If you are like me, the applet below will save you from the 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
Author David Wells wrote:
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
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 perceptibly sheds 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) the angles in "Morley's" triangles are distributed more or less evenly, with the smallest angle never below 35o. For smaller values of n, the borderline value is even higher: for n=4, the smallest angle is probably greater than 53o. Angles close to 53o occur only in very flat triangles which are beyond the reach of even a master-doodler.
For "more or less normal" triangles, the angles differ only slightly. I suspect that doodling might have led one to think that, for n=4, "Morley's" triangle is equilateral. That it is not was rigorously established in 1978 [Kleven]. No miracles here.
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:
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:
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.
Back to Morley's triangle. That this triangle is always equilateral is utterly startling. I have never heard a different 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
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
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.
On The Internet
Alex Bogomolny has started and still maintains a popular Web site Interactive Mathematics Miscellany and Puzzles to which he brought more than 10 years of college instruction and, at least as much, programming experience. He holds M.S. degree in Mathematics from the Moscow State University and Ph.D. in Applied Mathematics from the Hebrew University of Jerusalem. He can be reached at firstname.lastname@example.org
Copyright © 1997-1998 Alexander Bogomolny