Branko Grünbaum[1]

Cyclic ratio sums and products

The well known classical theorems of Menelaus and Ceva deal with certain properties of triangles by relating them to the products of three ratios of directed lengths of collinear segments.  Less well known is a theorem of Euler [2] which states, in the notation of Figure 1, that  ||QBj/AjBj|| = 1  for every triangle  T = [A1A2A3].  However, while the theorems of Menelaus and Ceva have been generalized to arbitrary polygons, and in many other ways –– see, for example, [4] [5] [6] –– until very recently there have been no analogous generalizations of Euler's result.  One explanation for this situation may be that attempts at straightforward generalizations lead to invalid statements.  An example of such a failed "theorem" is given by the question whether, in the notation of Figure 2,  ||QBj/AjBj|| equals 1 or some other constant independent of  Q  and the polygon.  Recently, Shephard [9] had the idea, apparently not considered previously, of attaching to the ratios  rj = ||(QBj)/(AjBj)||  certain weights  wj,  which depend on the polygon  p = [A1, A2, ... , An]  but not on the point  Q,  such that  wj rj = 1.  (In fact, Shephard established a much more general result in this spirit; its complete formulation would lead us too far from the present aims.)

By sheer chance, the same day I received from Shephard a preprint of [1], I happened to read [7], in which two different sums of ratios appear, one in Bradley's solution, the other in Konen's comments.  This coincidence lead me to consider whether these results could be generalized along Shephard's idea.  As it turns out, the answer is affirmative, and leads to a number of other results.

Let  p = [A1, A2, ... , An]  be an arbitrary  n-gon, and  Q  an arbitrary point, subject only to the condition that all the points  Bj  mentioned below are well determined.  On each side  AjAj+1  of  p  (understood as the unbounded line) the point  Bj  is the intersection with the line through  Q  parallel to  Aj+1Aj+2.  (Here, and throughout the present note, subscripts are understood  mod n).  This is illustrated in Figure 3 by an example with  n = 5.  We are interested in the ratios  rj = ||BjAj+1/AjAj+1||.  We denote by  D(UVW)  the signed area of the triangle  UVW  with respect to an arbitrary orientation of the plane and, more generally, by  D(p)  the signed area of any polygon  p,  calculated with appropriate multiplicities for the different parts if  p  has self-intersections.

Theorem 1.  For each polygon  p  we have  wj rj = D(p)  for all  Q,  where                  wj = D(AjAj+1Aj+2)  are weights that depend on the polygon  p  but are independent of the point  Q.

For a proof it is sufficient to note that

(i)        by straightforward calculations or by easy geometric arguments it can be shown that  rj = D(QAj+1Aj+2)/D(AjAj+1Aj+2);  and

(ii)       therefore the sum  wj rj  is equal to D(QAj+1Aj+2) = D(p),  since the triangles with vertex  Q  triangulate the polygon  p.

As a corollary we deduce at once that  wj sj  =  (wj ) D(p) , where             sj = ||AjBj/AjAj+1|| = 1 – rj.

In the special case that  p  is a regular  (n/d)-gon, all the weights  wj  are equal to the value  w = 4 sin3(dπ/n) cos(dπ/n).  (The regular  (n/d)-gon has  n  vertices and surrounds its center  d  times.  Successive vertices are obtained by rotation through 2πd/n,   see [1].  It is usually assumed that  n  and  d  are coprime, but this is a restriction that is unnecessary here and in most other contexts, and downright harmful in some cases, see, for example, [3]).  Hence, in this case one can divide throughout by  w,  and the result becomes

rj = D(p)/w = .                                                            (*)

Since the ratios  rj  involve only collinear lengths, the sum is invariant under affinities, and so the result  (*)  remains valid for all affine-regular  (n/d)-gons  p.  (An  (n/d)-gon is affine-regular if it is the image of a regular  (n/d)-gon under a nonsingular affinity.)  Thus in this special case we actually achieve the analogue of the generally invalid statement mentioned above.  Since all triangles are affine-regular, this establishes the condition for concurrency found by Václav Konen, mentioned in [7].  (We note that Shephard obtains in [9] the analogous generalization of Euler's result to affine-regular    n-gons.)  In the affine case, the above corollary can be simplified in the same way.  For       n = 3  this yields the condition for concurrency obtained by Bradley in [7].

From the above it follows that in the case of affine-regular polygons (but not for general polygons) we have

||BjCj/AjAj+1|| =  ,                                                 (**)

where the  Cj  is the intersection of the line  AjAj+1  with the parallel through  Q  to the line  Aj-1Aj  (see Figure 4).  For  n = 3  the right-hand side of (**) equals  1,  and the result coincides with Problem 16 in [8].

It may be observed that for  n = 3  and  d = 1  the right-hand side of condition  (*)  equals  1,  and the equality to  1  of the ratio sum is necessary and sufficient for the three parallels to the sides of the triangle to be concurrent, just as the equality to  1  of the product in Ceva's theorem for triangles is necessary and sufficient for theconcurrence of the Cevians.  However, for  n > 3  it is not obvious that the weights given above are the only ones which yield the right-hand constants for all  Q,  although one may conjecture that this is the case.  Naturally, for particular choices of  p  and  Q  other weights may be used.

The expression for  rj  obtained in (i), together with the analogous formula for the ratio  tj = ||AjCj/AjAj+1||  (in the notation of Figure 4) leads at once to the following:

Theorem 2.  For each polygon  p  with we have  = AjCj|| = 1  for all  Q.

Finally, since  QBjAj+1Cj+1  is a parallelogram for every  j,  we also have           ||BjQ/QCj+2|| = 1.

This last is a Ceva-type result which seems not to have been noticed previously.

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A referee's suggestions for improved presentation are acknowledged with thanks.

References

[1]       H. S. M. Coxeter, Introduction to Geometry. Wiley, New York 1969.

[2]       L. Euler, Geometrica et sphaerica quedam.  Mémoires de l'Académie des Sciences de St.-Petersbourg 5(1812), pp. 96 - 114. (This was submitted to the Academy on 1 May 1789; the actual publication date is 1815.) Reprinted in: L. Euleri Opera Omnia, Ser.1, vol. 26, pp. 344 - 358; Füssli, Basel 1953.

[3]       B. Grünbaum, Metamorphoses of polygons.  The Lighter Side of Mathematics, Proc. Eugène Strens Memorial Conference, R. K. Guy and R. E, Woodrow, eds. Math. Assoc. of America, Washington, D.C. 1994.  Pp. 35 - 48.

[4]       B. Grünbaum and G. C. Shephard, Ceva, Menelaus, and the area principle.  Math. Magazine 68(1995), 254 - 268.

[5]       B. Grünbaum and G. C. Shephard, A new Ceva-type theorem.  Math. Gazette 80(1996), 492 - 500.

[6]       B. Grünbaum and G. C. Shephard, Ceva, Menelaus and selftransversality.  Geometriae Dedicata 65(1997), 179 - 192.

[7]       H. Gülicher, Problem 1987.  Crux Math. 20(1994), p. 250.  Solution, ibid.  21(1995), pp. 283 - 285.

[8]       J. D. E. Konhauser, D. Velleman and S. Wagon, Which Way Did The Bicycle Go ?  Dolciani Math. Expositions No. 18.  Math. Assoc. of America, Washington, DC, 1996.

[9]       G. C. Shephard, Cyclic sums for polygons.  Preprint, August 1997.

University of Washington, Box 354350,

Seattle, WA 98195-4350

e-mail: grunbaum@math.washington.edu

Figure 1.  A theorem of Euler states that if  Bj  is the intersection of the line  AjQ  with the side of the triangle  A1A2A3  opposite to  Aj,  then  ||QBj/AjBj|| = 1.   Here and throughout the note,  ||MN/RS||  stands for the ratio of signed lengths of the collinear segments  MN  and RS.

Figure 2.  Attempts to generalize Euler's theorem in the form ||QBj/AjBj|| = const.  necessarily fail for  n > 3  (here  n = 5).  However, as shown by Shephard [9], it is possible to find weights  wj  which depend on the polygon but not on the position of  Q, such that  wj ||QBj/AjBj|| = 1.

Figure 3.  The point  Bj  is the intersection of the line  AjAj+1  with the parallel through  Q  to the line  Aj+1Aj+2.  The ratios  rj = ||BjAj+1/AjAj+1||  of directed segments are considered in Theorem 1.

Figure 4.  The point  Bj  is obtained as in Figure 3, while the point  Cj  is the intersection of the line  AjAj+1  with the parallel through  Q  to the line  Aj-1Aj.  The ratios  rj = ||BjAj+1/AjCj||  of directed segments are considered in Theorem 2.

[1]Research supported in part by NSF grant DMS-9300657.