Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

December 1998

Frank Morley will be remembered for a theorem he discovered before 1900 but did not enunciate, in print, until years later [Morley(1924, 1929)]. In a letter (August 1934) to Gino Loria [Loria], Morley wrote

The exact source is F.Morley in Mathematical Association of Japan for Secondary Mathematics, Vol. 6, Dec. 1924.

I mentioned the theorem also in "Extensions of Clifford's Chain Theorem", American Journal of Mathematics, Vol. 51, July 1929.

I mentioned the theorem to friends in Cambridge, England, around 1904, but as it was a part of a theory I did not make a paper of it. Thus it became public, and the Japanese reference is not I think the first reference in print.

For a short history of the theorem see [Oakley and Baker]. Early references did not mention Morley's name, and it appears that, as the theorem grew in popularity, Morley felt compelled to claim the authorship. But even then he could not bring himself to disassociate the theorem from the theory that led to it. His 1929's paper begins with the following paragraph

I propose to state more fully what was implied in the final section of my paper "metric geometry of the plane n-line," Transactions of the American Mathematical Society, Vol. 1 (1900), p. 115. I refer to this as M.G. First, let us recall Clifford's Theorem, Works, p. 51.

The theorem is informally articulated in a single paragraph in the middle of the paper. Why this reluctance to state a surprising result apart from the underlying theory? Mathematicians thrive on announcements of newly proven results. Proofs and theories undergo continuous metamorphosis: some are being simplified, others generalized, yet others are found to apply to unexpected fields - examples abound. So usually a good theorem deserves attention by virtue of its own content, independent of a particular proof or a theory that led to it.

Morley's research started with the incidence results of Steiner, Kantor and Clifford that in themselves are among the most intriguing in geometry. Not only his method led to powerful generalizations, it did this in a most elegant way. Morley's been thorough. He applied his method to circumscribed as well inscribed circles, he studied incidence of their centers as well as their intersection points with other circles, he moved from circles to higher order curves. He might have been feeling himself sitting on a treasure trove full of mathematical wonders. One of these wonders - an accidental consequence of a more general result - felt a little out of place. Was he thinking that taken out of context, the theorem would lose much of its charm and motivation? Or, perhaps, he did not want to divert attention from his other results? We can't say now. But let's have a look at a sample of Morley's handiwork.

First of all, the object of his study was the plane n-line, i.e. a set of n lines in a plane. The term n-line underscores our interest in the properties the n lines may have en ensemble. The best studied case is of course n = 3. In general, three lines intersect at three points. There is a circle incident to all three points (the circumcircle). The incircle and 3 excircles each touch all three lines. The incenter is incident to each of the three lines joining a vertex with the opposite excenter. There is the orthocenter, etc.

A 4-line contains four 3-lines. The 4 circumcircles meet at a point (Miquel's point) and their centers lie on a circle (Steiner's circle). From a 5-line we have five 4-lines; the centers of their Steiner circles lie on a circle (Kantor's circle.) The general term for all those circles is a center-circle. From a 6-line we have six 5-lines and 6 center-circles whose centers lie on a circle.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

On the other hand, 5 Miquel points of a 5-line lie on a circle (Clifford's circle), 6 Clifford's circles of a 6-line intersect at a point (Clifford's point.) 7 Clifford's points of a 7-line lie on Clifford's circle. 8 Clifford circles of an 8-line intersect at Clifford's point, and so on ad infinitum. The whole configuration is known as Clifford's chain.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

Back to the center-circles that intersect at a point (say P) and whose centers lie on a circle - the next center-circle in the hierarchy of circles. The point P is also incident to this latter center-circle. This is how the cardioids came into the play - as envelopes of families of circles. Not surprisingly, the common point of the center-circles lies at the cusp of the cardioids.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

This was just the beginning. Morley's theory evolved over the space of three papers [Morley (1900, 1903, 1907)]. Only in the second one he used the now standard theorem-proof style. The number of theorems in that paper is 10. There are that many statements in the other two articles. I'll return to Morley's Trisector Theorem on another page, but, for now, let's turn to the algebra of Morley's method.

Yes, with all the geometric context of his theory, Morley's method is purely algebraic, founded on the theory of complex numbers. Curves are mappings x = f(t) from the unit circle. x = X + iY, where X and Y are the plane Cartesian coordinates. y = X - iY is the conjugate of x, and together x,y are called the circular coordinates in the plane. The short term for the complex numbers t on the unit circle is turn. Circles in these notations are given by the linear equation:


x = a1 - a2t,

a1 being the center and |a2| the radius of the circle. A (straight) line is defined by a point - reflection of the origin in that line. The perpendicular bisector of the segment [0,x1] is given by the following equation


x = x1t1/(t1 - t),

where t1 is an arbitrary turn. (This is a Möbius transform that takes t1 to infinity and 0 into x1.) t1 is selected in a peculiar manner as


t1 = y1/x1.

With this choice of t1, (2) becomes


xt1 + y = x1t1.

Two lines

xt1 + y = x1t1
xt2 + y = x2t2.

meet at the point

x12 = x1t1/(t1 - t2) + x2t2/(t2 - t1).

For three lines defined by x1, x2, and x3, consider the circle


For t = t3, the right-hand side in (5) becomes x12. Therefore the circle passes through the point of intersection of lines #1 and #2 and, similarly, through the other two intersection points of the 3-line. This is thus the circumcircle of the 3-line. Its center a1 and radius |a2| are given by


By induction, (5) generalizes to


which is the center-circle of an n-line with the center a1 and radius |a2| determined from


This is obviously so because fn(tk) reduces to a1 evaluated for n-1 points x1, ..., xk-1, xk+1, ..., xn.

Formula (5') for the center-circle of the n-line has grace and clarity that are unlikely to benefit from graphical illustration. The natural limitations of the applet at the top of the page only underscore the generality inherent in (5'). With the advent of new technology, we became better doodlers. We anticipate that new hands-on experiences will help fire up students' imagination and make math more palatable. Morley's theory serves to remind us that at least some appreciation of mathematics may be gained in the old-fashioned way - through the study and understanding of its language.

Morley goes on to define characteristic constants of an n-line.


where, for simplicity, the dependency on the number of lines n is only implicit. The characteristic constants of the (n-1)-line obtained from a given n-line by omitting the line x1 are successively equal to

a1 - a2t1, a2 - a3t1, ..., an-1 - ant1.

If two lines x1 and x2 are omitted, the characteristic constants of the remaining (n-2)-line become

a1 - a2(t1 + t2) + a3t1t2, a2 - a3(t1 + t2) + a4t1t2, ...

Consider the equation:

x = a1 - a2(t1 + t2) + a3t1t2

This is the center of the center-circle of the (n-2)-line obtained by omitting lines x1 and x2. Replace t2 with t:

x = a1 - a2(t1 + t) + a3t1t

This is the equation of the center-circle of the (n-1)-line obtained by omitting x1. Let's show that n circles


x = a1 - a2(tk + t) + a3tkt, k = 1, 2, ..., n

all meet at a point. When a2 = 0, they are all the same circle. Assume, a2≠0 and let x = a1 - a2b3/b2. (Constants bi are respective conjugates of the constants ai. Thus ai, bi form a pair of circular coordinates.) (8) then becomes

0 = b3/b2 - (tk + t) + tkta3/a2, k = 1, 2, ..., n

which, when solved for t, yields a turn - a number from the unit circle. Therefore, all circles (8) pass through the point x = a1 - a2b3/b2.


  1. F.Morley, On the Metric Geometry of the N-Line, Trans Amer Math Soc, 1 (1900) 97-115.
  2. F.Morley, Orthocentric Properties of the Plane n-Line, Trans Amer Math Soc, 4 (1903) 1-12.
  3. F.Morley, On Reflexive Geometry, Trans Amer Math Soc, 8 (1907) 14-24.
  4. F.Morley, in Mathematical Association of Japan for Secondary Mathematics, v 6, Dec 1924.
  5. F.Morley, Extensions of Clifford's Chain-Theorem, Amer J Math, 51 (1929) 465-472.
  6. G.Loria, in Mathematical Gazette, 23 (1939) 364-372.
  7. C.O.Oakley and J.C.Baker, The Morley Trisector Theorem, Amer Math Monthly, 85 (1978) 737-745.

Morley's Miracle

On Morley and his theorem

  1. Doodling and Miracles
  2. Morley's Pursuit of Incidence
  3. Lines, Circles and Beyond
  4. On Motivation and Understanding
  5. Of Looking and Seeing

Backward proofs

  1. J.Conway's proof
  2. D. J. Newman's proof
  3. B. Bollobás' proof
  4. G. Zsolt Kiss' proof
  5. Backward Proof by B. Stonebridge
  6. Morley's Equilaterals, Spiridon A. Kuruklis' proof
  7. J. Arioni's Proof of Morley's Theorem

Trigonometric proofs

  1. Bankoff's proof
  2. B. Bollobás' trigonometric proof
  3. Proof by R. J. Webster
  4. A Vector-based Proof of Morley's Trisector Theorem
  5. L. Giugiuc's Proof of Morley's Theorem
  6. Dijkstra's Proof of Morley's Theorem

Synthetic proofs

  1. Another proof
  2. Nikos Dergiades' proof
  3. M. T. Naraniengar's proof
  4. An Unexpected Variant
  5. Proof by B. Stonebridge and B. Millar
  6. Proof by B. Stonebridge
  7. Proof by Roger Smyth
  8. Proof by H. D. Grossman
  9. Proof by H. Shutrick
  10. Original Taylor and Marr's Proof of Morley's Theorem
  11. Taylor and Marr's Proof - R. A. Johnson's Version
  12. Morley's Theorem: Second Proof by Roger Smyth
  13. Proof by A. Robson

Algebraic proofs

  1. Morley's Redux and More, Alain Connes' proof

Invalid proofs

  1. Bankoff's conundrum
  2. Proof by Nolan L Aljaddou
  3. Morley's Theorem: A Proof That Needs Fixing

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