Cut The Knot!
An interactive column using Java appletsby Alex Bogomolny
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Cut The Knot!An interactive column using Java appletsby Alex Bogomolny |
December 1998
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Frank Morley will be remembered for a theorem (discussed in last month's column) which 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
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
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 did his method lead to powerful generalizations, it did this in a most elegant way. Morley was 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 felt 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, that 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 apposite 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. 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. 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 was just the beginning. Morley's theory evolved over the space of three papers [Morley (1990, 1903, 1907)]. Only in the second one did he use 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:
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
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
With this choice of t1, (2) becomes
Two lines
meet at the point
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 If two lines x1 and x2 are omitted, the characteristic constants of the remaining (n-2)-line become Consider the equation: This is the center of the center-circle of the (n-2)-line obtained by omitting lines x1 and x2. Replace t2 with t:
This is the equation of the center-circle of the (n-1)-line obtained by omitting x1. Let's show that n circles
all meet at a point. When a2 = 0, they are all the same
circle. Assume, a2
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. References
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 alexb@cut-the-knot.com Copyright © 1997-1998 Alexander Bogomolny |
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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