Problem
ALG-B2-M08-P005 Squares of sides
#5
★★★☆☆ Level 3 of 5
If \(a,b,c\) are triangle sides, prove \(a^2+b^2+c^2<2(ab+bc+ca)\).
Hint. Use \(a=y+z\), \(b=z+x\), \(c=x+y\).
After substitution, \(2(ab+bc+ca)-(a^2+b^2+c^2)=4(xy+yz+zx)>0\). Hence the strict inequality holds.
Teaching goal: recognize the appropriate substitution and check the domain.