Cut The Knot!An interactive column using Java applets
by Alex Bogomolny
This is the fourth in the series of columns (see links at the bottom of the page) devoted to Frank Morley's work. It's time to deal with what amounts to his proof of the Trisector Theorem. I'll outline Morley's reasoning first and then supply necessary details.
The second section of Morley's  paper begins with the following paragraph:
For a 3-line there exist 4 circles that touch all three lines. For a 4-line there are 8 cardioids that touch all 4 lines. Increasing the number of cusps, curves are produced that touch all n +1 lines for greater n's.
Morley first found a general formula for the incenters in terms of the reflections of the origin in each of the lines. By letting one of the lines vary, he got the locus of incenters of the curves that touch the remaining n lines.
For example, if n = 2, we are talking of circles that touch three lines. With one line removed (or varying), the focus is on the circles that touch the remaining two lines. The locus of their centers is the union of two angle bisectors - two one line sets forming an angle of /2. These are the axes of the 2-line.
For n = 3, we have to consider two cases. When the remaining 3 lines are in general position the locus of the incenters of the cardioids that touch all three lines is a union of three sets of three parallel lines forming angles of /3 (and, naturally, 2/3.) These are the axes of the 3-line. Naturally, the axes form equilateral triangles - 27 of them. An important observation is that the vertices of these triangles - the points where two axes meet - have the property that the corresponding cardioid touches one of the given lines (of the 3-line) twice.
For larger n, we get nn-2 sets of n parallel lines forming angles that are multiples of /n. On the whole, there are nn-1 lines called axes of the n-line. Pick one line from each family. These lines form an equiangular n-gon. n = 3 is a very special case in that every equiangular 3-gon is also equilateral. This is how Morley arrived at the equilateral triangles.
Oakley and Baker stated in 1978
As we just saw, this is a clear understatement. Morley's theory directly leads to all the possible equilateral triangles. If anything, when they emerged in his mind's eye, they emerged in a mesh of parallel lines, all 27 of them. (The mistaken count of 18 had probably originated with W.J.Dobbs who, in 1938, came up with elementary trigonometric proofs for 18 of the 27 triangles. He listed also 9 non-equilateral "Morley" triangles and thanked S.W.Finn for pointing out the missed 9 equilateral ones that he refused to designate as "Morley". Unfortunately, there was no definition of what qualifies a triangle as "Morley".)
For n = 3 there is still another case to consider. The three lines may be concurrent. Furthermore, the most relevant to the theorem is the case where two of the lines coincide. In this case, we look at the cardioids that touch 2 lines of which 1 is touched twice. The centers of such cardioids are located on the trisectors of the angles formed by two lines. In this case, the trisectors serve as the axes of the 3-line similarly to the case of n = 2 where the role of the axes was played by the angle bisectors.
Morley's theorem is a combination of the two cases. The set of all cardioids that touch the 3-line falls into 32 discrete systems specified by the axes. For every axis, cardioids touch the three lines in a peculiar manner. Transition between the systems occurs at the points of intersection of the axes. Cardioids with centers at those points touch one of the lines twice. Their centers thus lie on the trisectors of the angles adjacent to that line.
The applet illustrates the background of Morley's theory. You can display the axes and the angle trisectors. Complex turns corresponding to the trisectors arise as solutions to a cubic equation that does not distinguish between primitive roots of unity. Therefore, along with an angle A, we should consider angles A + 2 and A + 4. The lines thus obtained are designated as spurious although they play as important a role as the angle trisectors.
The vertices of the given triangle are draggable. To test the theory, display a cardioid that, besides being draggable itself, can be modified by dragging either its center or the cusp.
Morley's  paper makes the situation quite transparent. He changes notations a little bit. A straight line is defined by the reflection of the origin in it. Let the point be xi. Then the line can be written as either xti + y = xiti (as in the paper ) or x + yti = xi. (Recollect that x and y form a pair of circular coordinates.) In the first case, ti = yi/xi; in the second, ti = xi/yi. We use the latter form.
is a (self-conjugate) equation of the straight line that is the perpendicular bisector of the segment joining S with the origin. If T is the conjugate of S, then A = S/T, A is a turn - a complex number on the unit circle. A is called the clinant of the line. The equation (1) is convenient in that it immediately shows the point S associated with the line. Let's register two important facts:
A curve can be described by its map- or point-equation as a function of a turn, or by its line-equation as the envelope of a family of lines. The line-equation of a cardioid with the center at x0 is given by
where the coefficients a and b are conjugate. For a given t, the clinant of the line tangent to the cardioid is t3. Choose four turns t1, t2, t3, t4. The four corresponding tangents are defined by the reflections of the origin x1, x2, x3, x4. We do not need explicit expressions but only note that the clinants of the tangents are also given by xi/yi, I = 1, 2, 3, 4. Thus we have
We are going to express x0 in terms of xi's and ti's. Setting x = 0 and t = ti in (2) gives
Eliminate x0, a, and b we have a determinant
Appearance of the Vandermonde determinants may be at first surprising. But the fact is that, by and large, Morley's was the theory of complex interpolation in a geometric disguise. (For example, the circumcircle interpolates through the vertices of a triangle.)
Taking into account (3) we simplify (5) to
The conjugate of (5) is
From (6) and (7) we can form a combination x0 + y0t1t2t3. We immediately see that in this combination the term with x4 cancels out. Expanding the determinants with respect to the fourth column, we also observe that the final expression does not contain t4 either. Thus the result is a self-conjugate (because of (3)) equation in the form (1)
This is the locus of centers of cardioids that touch the three lines defined by x1, x2, and x3. (8) seems to be an equation of a straight line, but this is not so. The reason is that the turns ti are not determined uniquely. From (3) we are only given their cubes. Thus each is only defined up to a primitive root of unity. Letting ti's change in (8), we see that the locus of the incenters comprises a family of straight lines. There are only 3 different clinants and, as implied by the form of the right-hand side of (8), 9 different lines. So we already know that there are three families of three parallel lines each.
To determine the angles between them observe that the clinant t1t2t3 of (8) is the geometric mean of the clinants ti3 of the three lines, see (3). Denote the angles formed by x1, x2, and x3 and (8) with a base line as a1, a2, a3, and a, respectively. Then (refer to the facts we registered at the beginning) a is the arithmetic mean of a1, a2, a3 modulo . This is where the multiples of 60o come from.
Finally observe that, if the reasoning of the last paragraph is applied to the case of three concurrent lines of which two coincide, we get, say,
which points to a trisector of the two lines. Actually (8) now yields three of them. To get the other three consider a = (2a1 + a2)/3 (mod ). We can still use (8) letting, say, t3 = t2, where is a primitive root of unity, 3 = 1. It is perfectly straightforward to verify that the three axes thus obtained are concurrent. (A word of advice: Tackle the left-hand sides of (8) and use the criterion x1(t2 - t3) + x2(t3 - t1) + x3(t1 - t2) = 0 for the concurrency of three lines.) Their common point coincides with the point of intersection of the two given lines.
Morley of course developed his theory for a general n-line. He got the following table:
In his own words:
I realize that this is highly unfair to compare the edifice erected by Morley with the ad hoc cabins of proofs of what is now known as his Trisector theorem. Taken out of context, the theorem indeed becomes a miracle that inspires admiration but loses the sparkle of intrinsic consistency. To have a fighting chance to a claim for motivation, an elementary proof should probably handle the whole set of 27 triangles simultaneously, which means proving existence of three sets of parallel lines cutting at the 60o angles. But even then an elementary framework will not attain the broad outlook of Morley's theory that includes the angle bisectors and trisectors (and more) under a single roof.
So this is the end of an endeavor to understand Frank Morley's work and particularly his Trisector theorem. I began writing this column with a quiet satisfaction that, after diligently plowing through Morley's papers, I finally reached a stage where it became possible to assert my inderstanding of Morley's theorem. At this point, I completed the first half of the column, inserted the applet and ran my browser to see how things looked so far. This presented me with the first opportunity to play with the applet. For, while writing it, I did not really experiment with it. Just made sure there were no bugs.
Toying with the applet gave me an additional insight. The axes of a 3-line serve as a sort of index into how a cardioid may touch the three lines. (I remove my hat to Frank Morley who saw everything so clearly without the benefit of modern day gadgets.) Well, you can say, is it not what you wrote above, when outlining Morley's reasoning in the first place? Yes and no. Is it internalization that I lacked before? Probably. Whatever it is called, this something had clearly enhanced my understanding of the theorem. The meaning of it may be different from what Donald Newman meant when he asked "Were we to give up, forever, understanding the Morley Miracle?" But I certainly did not get it upon reading his and other proofs.
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 email@example.com
Copyright © 1997-1998 Alexander Bogomolny