Denote the 3 Cirles are (A; a), (C; c), (B; b) in the order given. Projection A'B' of AB on the tangent line is sqrt[(a+b)^2 -(a-b)^2] = 2sqrt(ab) Similar expressions for Proj's A'C', B'C' of AC, BC on the tangent line. Clearly A'B' = A'C' +B'C' implies 2sqrt(ab) = 2sqrt(ac) + 2sqrt(bc) Divide thro'out by 2sqrt(abc) to get the required result.
Denote the 3 Cirles are (A; a), (C; c), (B; b) in the order given.
ReplyDeleteProjection A'B' of AB on the tangent line is sqrt[(a+b)^2 -(a-b)^2] = 2sqrt(ab)
Similar expressions for Proj's A'C', B'C' of AC, BC on the tangent line.
Clearly A'B' = A'C' +B'C' implies
2sqrt(ab) = 2sqrt(ac) + 2sqrt(bc)
Divide thro'out by 2sqrt(abc) to get the required result.