Proof

cosA+cosC=2(1-cosB)

=> 2cos((A+C)/2)cos((A-C)/2) = 4sin²(B/2)
=> 2sin(B/2)cos((A-C)/2) = 4sin²(B/2)
=> cos((A-C)/2) = 2sin(B/2)
=> cos((A-C)/2) =2cos((A+C)/2)
=> cos((A-C)/2) - cos((A+C)/2) = cos((A+C)/2)
=> 2sin(A/2)sin(C/2) = sin(B/2)

=> 2√(s-b)(s-c)√bc√(s-a)(s-b)√ab = √(s-a)(s-c)√ac

=> 2(s-b) = b
=> a+b+c-2b = b
=> a+c=2b (proved)