Let ABCD be a cyclic quadrilateral with centre O. Let the lines AB and CD meet at P, AD and BC meet at Q, AC and BD meet at R. Let OP meet QR at X, OQ and PR meet at Y, OR and PQ meet at Z. It transpires that there are no fewer than 37 generalised circles passing through four points, namely: (ABCD) (*ABP) (*CDP) (*ACR) (*BDR) (*ADQ) (*BCQ) (*OPX) (*QRX) (*OQY) (*PRY) (*ORZ) (*PRZ) (OPYZ) (OQXZ) (ORXY) (PQXY) (PRXZ) (QRYZ) (OYBC) (OYAD) (OXAB) (OXCD) (OZAC) (OZBD) (BCXR) (BCZP) (ADXR) (ADZP) (ABYR) (ABZQ) (CDYR) (CDZQ) (ACXQ) (ACYP) (BDXQ) (BDYP) The asterisk (*) indicates the point at infinity on the Riemann sphere. The diagram has 192 automorphisms, containing a subgroup of eight undirected-angle-preserving mappings, itself divided into four Möbius transformations and four inversions (about O, P, Q and R). All 37 generalised circles can be obtained by the following systematic procedure: 1. Draw the following 4*4 matrix: ABDC ABDC OPQR *XYZ 2. Optionally remove either the top- or bottom-half of the matrix. In this example, I choose to omit this step: ABDC ABDC OPQR *XYZ 3. Optionally remove any two columns of the matrix. In this example, I remove the second and fourth columns: AD AD OQ *Y 4. Choose any four points, such that the number of selected points in each row is the same (4/number of rows), and the number of selected points in each column is the same (4/number of columns). In this example, there must be two selected points in each column and one in each row: (A)D A(D) (O)Q *(Y) 5. The selected points are necessarily concyclic. In this example, (ADOY) are concyclic. The 192 automorphisms of the diagram can be identified with compositions of the following operations on the 4*4 matrix: I. The four columns can be freely permuted. Möbius tranformations correspond to permutations with the cycle structure (**)(**). II. The bottom two rows can be interchanged. This is equivalent to inversion about O. III. The four columns of the lower half-matrix can be permuted with cycle structure (**)(**). It is clear that I, II and III are subgroups of 24, 2 and 4 elements, respectively, and they together generate all 192 elements of the group. However, there is another operation on the matrix that does not correspond to an automorphism of the 37-circle diagram: interchanging the top- and bottom-halves of the matrix. This is due to the asymmetry of having four points duplicated in the matrix. I believe that Ivan's 37- circle configuration is actually a degenerate case of a more general configuration of 16 points. Does such a configuration exist? Sincerely, Adam P. Goucher