My method is similiar to sir bhatt

but still as you need diff method i think i might post it

by cauchhy swartz inequality

(a+b+c)^2<=(sum)((aa+bb)/a+(bb+cc)/b+(cc+aa)/c)

1<=s(1+bb/a+cc/b+aa/c)

let a<b<c

a^2<b^2<c^2

1/c<1/b<1/a

min value by rearrange.inequality

a+b+c=1

1/(1+1)<=s

s>=1/2

 

sorry i didnt see that you dont want sol. through re. inequality.