Let loga / (b – c) = logb / (c – a) = logc / (a – b) = k
loga = k (b – c) , a = 10 k (b – c) logb = k (c – a) , b = 10 k (c – a)
logc = k (a – b) , c = 10 k (a – b)
a(b – c) = {10 k (b – c)}^(b – c) = 10 k (b – c)^2
b(c – a) = {10 k (c – a)}^(c – a) = 10 k (c – a)^2
c(a – b) = {10 k (a – b)}^(a – b) = 10 k (a – b)^2
a(b – c) + b(c – a) + c(a – b) = 10 k (b – c)^2 + 10 k (c – a)^2 + 10 k (a – b)^2 ….........(1)
From (1), the minimum value of a(b – c) + b(c – a) + c(a – b) occurs when a = b = c which is 3 (you can check).
a(b – c) + b(c – a) + c(a – b) 
3 a(a)+ b(b) + c(c) = 10 k (b – c) a + 10 k (c – a) b + 10 k (a – b) c ….............(2)
From (2), the minimum value of a(a)+ b(b) + c(c) occurs when a = b = c again which is 3.
a(a)+ b(b) + c(c) 3 
