... planeABC, the points X, Y, Z, W lie on the cross-section of the sphere,which is a circle. Therefore BY · BX = BZ · BW by power-of-a-point; this clearly implies AB = BC, and so the center of ... there are finitely many n-tuples, some n-tuple mustrepeat, and the first such must be the original configuration, sinceeach n-tuple (y1, . . . , yn) comes from a unique n-tuple, namely (yn−y1− ... Sithat of the i-th smallpyramid, and kithe coefficient of similarity between the i-th pyramidand π. We concludeS =5i=1Si2≤ 55i=1Siby the power mean (or Cauchy-Schwarz) inequality....