for a,b,c real positive numbers such that a+b+c=2, a<b+c, b< c+a , c< a+b; show that

a^2 + b^2 + c^2 + 2abc < 2 .

first, i try to find abc and i got

abc< abc+a(c^2)+(a^2)b+c(b^2)+(c^2)b+ a(b^2)+abc

then i find (a^2)+(b^2)+(c^2)< 2(a^2)+2(b^2)+2(c^2)+2(ab+ac+bc)

i add both 2abc+(a^2)+(b^2)+(c^2) but i didnt end up with proved..

what can i do?