AN ABSOLUTE PROPERTY OF FOUR MUTUALLY TANGENT CIRCLES H.S.M. Coxeter University of Toronto Toronto M5S 3G3, Canada ABSTRACT When Bolyai János was forty years old, Philip Beecroft discovered that any tetrad of mutually tangent circles determines a complementary tetrad such that each circle of either tetrad intersects three circles of the other tetrad orthogonally. By careful examination of a new proof of this theorem, one can see that it is absolute in Bolyai s sense. Beecroft s double-four of circles is seen to resemble Schläfli s double-six of lines. 1. INTRODUCTION The absolute property of four mutually tangent circles that I am describing seems to have been discovered by Mr. Philip Beecroft (of Hyde Academy, Cheshire, England) and published in The Lady s and Gentleman s Diary for the year of our Lord 1842, being the second after Bissextile, designed principally for the amusement and instruction of Students in Mathematics: comprising many useful and entertaining particulars, interesting to all persons engaged in that delightful pursuit. [Beecroft, p. 92]. In Beecroft s own words, If any four circles be described to touch each other mutually, another set of four circles of mutual contact may be described whose points of contact shall coincide with those of the first four. I like to name this Beecroft s theorem and to express it as follows. 2. BEECROFT S THEOREM Four circles, mutually tangent at six distinct points, determine four other circles, mutually tangent at the same six points, such that each circle of either tetrad intersects three circles of the other orthogonally at the points of mutual contact of those three. 1
b 3 a 1 b 4 b 2 a 3 a 2 a 4 b 1 FIGURE 1. Beecroft s double-four of circles In Figure 1 we see four mutually tangent circles a 1, a 2, a3, a 4 (dark) and another such set of four b 1, b2, b3, b4 (light), such that b 1 passes through the points of mutual contact of a, a, a 4, and b 2 3 2 through the points of mutual contact of a, a, 1 3 a 4 and so on. In other words, am and bn intersect each other orthogonally whenever m! n. This figure makes the theorem almost obvious, but for the sake of completeness it seems desirable to consider further details. 2
FIGURE 2. How a, a, a 4, 2 3 determine 1 b 3. A NEW PROOF How do we know that the common tangents of the three circles touching one another are concurrent? [Coxeter 2, pp. 311, 316]. It is because these common tangents are radical aces of pairs and all pass through the radical center of these three circles. These three tangents, drawn from the radical center to the points of contact, all have the same length and thus are radii of a new circle intersecting each of the three circles orthogonally (see Figure 2). 3
Since both b 1 and b 2 intersect a 3 and a 4 at their point of contact, the four b- circles yield the four a-circles by the same procedure that led from the a-circles to the b- circles. Since no step in this proof uses Euclid s parallel postulate, directly or indirectly, Beecroft s theorem is indeed an absolute property of four mutually tangent circles; it holds not only in the Euclidean plane but also on a sphere and in the hyperbolic plane [Carslaw, pp. 27-32]. In the hyperbolic case one or more of the four circles may be replaced by a horocycle or a hypercycle (i.e., an equidistant curve). But Poincaré s circular model for the hyperbolic plane rules out the possibility of four mutually tangent horocycles! 4. BEECROFT S THEOREM ON A SPHERE A spherical version of Beecroft s theorem is provided by two tetrads of mutually tangent circles which lie on the in-sphere of a cube and are the in-circles of the faces of two regular tetrahedra inscribed in the cube. This compound of two tetrahedra is often called stella octangula [Coxeter 1, p. 158; 3, p. 166]. The face-centres of the cube, which are the common midpoints of pairs of crossing edges of the two tetrahedra, are the six points at which corresponding circles of Beecroft s two tetrads intersect orthogonally. Reciprocation with respect to the sphere transforms the vertices of each tetrahedron into the face-planes of the other. Thus corresponding edges are polar lines. In terms of Cartesian coordinates, the eight vertices of the cube are naturally taken to be ( ± 1, ± 1, ± 1), with an even number of minus signs for one tetrahedron, an odd number for the other. The six face-planes of the cube have the equations 4
x = ±1, y = ±1, z = ±1, and the eight face-planes of the two regular tetrahedra are ± x ± y ± z = 1 with an odd number of minus signs for one tetrahedron, whose vertices include ( 1,1,1 ) in the plane x + y! z and an even number of minus sings in the other, whose vertices include (! 1,! 1,! 1) in the plane = 1 x! y! z = 1. In other words, Beecroft s 4 + 4 circles are the sections of the sphere x 2 2 2 + y + z = 1 by those eight planes. 5. THE DOUBLE-FOUR OF CIRCLES AND THE DOUBLE-SIX OF LINES It is, perhaps, not too fanciful to recognize some analogy relating Beecroft s doublefour of circles & a1, a 2, a3, a 4# $! 1 2 3 4 % b, b, b, b " in the plane, and Schläfli s double-six of lines & a1, a 2, a3, a 4, a5, a $ 1 2 3 4 5 6 % b, b, b, b, b, b in the projective space [Schläfli 2, p. 213]. (Schläfli was a Swiss contemporary of Bolyai and Beecroft.) #! " 6 5
In Beecroft s double-four, two circles am and bn intersect orthogonaly whenever m! n. In Schläfli s double-six, two lines am and bn meet whenever m! n. 6. SCHLÄFLI S THEOREM In Schläfli s own words [Schläfli 2, p. 214] (slightly altered because he abandoned his a mbn notation in favour of A, B, C, D, E, a, b, c, d, e, f ). The double-sixes give rise to the remark that there is here exposed to view an apparently very elementary theorem which may be thus enunciated: Draw at pleasure five lines a, a, 2 3, a 4, a5 a6 which meet just one line b 1. Then (since any four mutually skew lines usually have just two transversals), any four of the five lines may be intersected by another line besides b 1. In this way we have the five tetrads intersected by a a 4a a 5 6, a a a, 3 2 4a5 6 a a3a a 5 6, a, 2 3a a 2 4 6 a a a a 2 3a 4 5 b 2 b 3 b 4 b 5 b 6 respectively. The apparently elementary theorem states that the five lines b, b, 2 3, b4, b5 b6 have a transversal, which we naturally name 1 a, this completing the double-six. Is there, for this elementary theorem, a demonstration more simple than the one derived from the theory of cubic forms? Schläfli s challenging question has been answered by a number of geometers, as one can see in the list of References. 6
REFERENCES [1] H.F. Baker, A geometrical proof of the theorem of a double six of straight lines. Proc. Royal Soc. A 84 (1911), p. 597. [2] H.F. Baker, The General Cubic Surface, Principles of Geometry, Vol 3, Solid Geometry, Cambridge University Press (1934), pp. 159 and 225. [3] P. Beecroft, The Concordent Circles. The Lady s and Gentleman s Diary, The Company of Stationers, London (1843). [4] H.S. Carslaw, The Elements of Non-Euclidean Plane Geometry and Trigonometry, Longmans, London (1916). [5] H.S.M. Coxeter, Introduction to Geometry (2 nd ed.), Wiley, New York (1969). [6] H.S.M. Coxeter, Inversive Geometry, in Educational Studies in Mathematics, Vol 3 (1971), pp. 310-321. [7] H.S.M. Coxeter, A Geometriák Alapjai, Müszaki Könyvkiadó, Budapest (1973). [8] Harold L. Dorwart, The Schläfli Double-Six Configurations, C.R. Math Rep. Acad. Sci. Canada, Vol 15 (1993) pp. 54-58. [9] John Dougall, The Double-Six of Lines and a Theorem, in Euclidean Plane Geometry, Proc. Glasgow Math. Assoc., Vol 1 (1952), pp. 1-7. [10] Asijiro Ichida, A Simple Proof of the Double-Six Theorem, Tohuku Math. Journ., Vol 32 (1929) pp. 52-53. [11] R. J. Lyons, A Proof of the Theorem of the Double-Six, Proc. Cambridge Philos. Society, Vol. 37 (1941) pp. 433-434. [12] L. Schläfli, Theorie der vielfachen Kontinuität, Gesammelte Mathematische Abhandlungen, Band I, Verlag Birhäuser, Basel (1953). 7
[13] L. Schläfli, An attempt to determine the twenty-seven lines upon a surface of the third order, and to divide such surfaces into species in reference to the reality of the lines upon the surface, Gesammelte Mathematische Abhandlungen, Band II, Verlag Birhäuser, Basel (1953). [14] B. Segre, Sulla costruzione delle bisestuple di nette, Rend. Acad. Naz. Lincei (6) Vol II (1930), pp. 448-449. [15] J.A. Todd, Proc. Lon. Math. Soc. 9 (1911), p.178. [16] J.A. Todd, Proc. Camb. Phil Soc. 26 (1930), p. 332. [17] C. Yamashita, An elementary and purely synthetic proof for the double-six theorem of Schläfli, Tohoku Math. Journ. (2) Vol 5 (1954), pp. 215-219. 8