... so thatx 2 + b 2 = y 2 + a 2 and (x 2 − y 2 ) 2 + c 2 = y 2 + a 2 .Leta 2 − b 2 = λ and a 2 − c 2 = µ, i.e., x 2 − y 2 = λ and x 2 − 2xy = µ.From the second equation we deduce that 2y = x −µx. ... x 2 + y 2 , (y − a) 2 + z 2 and (x − a) 2 + (z − a) 2 ,respectively. All these numbers cannot be simultaneously s maller than1 2 a 2 becausex 2 + (x −a) 2 ≥a 2 2, y 2 + (y − a) 2 ≥a 2 2and ... are(x, y, z), thenAM 2 BM 2 =(x + a) 2 + y 2 + z 2 (x −a) 2 + y 2 + z 2 .The equation AM : BM = k reduces to the formx +1 + k 2 1 −k 2 a 2 + y 2 + z 2 =2ka1 −k 2 2 .This equation...