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.

* * * * *

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.