((a + c)/a)(a/b) = 2
a + c = 2b ... (i)
⇒ a(1 – x2) + 2bx + c(1 + x2) = 0
x2(c – a) + 2bx + c + a = 0
for equal roots b2 – c2 + a2 = 0
a2 + b2 = c2 ... (ii)
Hence it is right angle at ∠C
Putting value in equation (2) from equation (1)
⇒ a2 + b2 = (2b – a)2 ⇒ a2 + b2 = 4b2 + a2 – 4ab
⇒ 3b2 = 4ab ⇒ b = 4/3a; c = 5/4b ⇒ c = 5/3a
Hence a = 3k, b = 4k, c = 5k
sinA + sinB + sinC = 3/5 + 4/5 + 1 = 12/5