Consider 4 circles S1, S2, S3, S4. Suppose that S1, S2 intersect at A1 and A2, circles S2, S3 intersect at Bi and B2, circles S3, S4 intersect at Ci and C2, and circles S4, S1 intersect at D1 and D2....

Extracted text: Consider 4 circles S1, S2, S3, S4. Suppose that S1, S2 intersect at A1 and A2, circles S2, S3 intersect at Bi and B2, circles S3, S4 intersect at Ci and C2, and circles S4, S1 intersect at D1 and D2. Suppose furthermore that the points A1, Prove that the points A2, B2, C2, D2 B1, C1, D1 are on a circle S (or on a line) are also on a circle (or on a line).