Problem
ALG-B1-M06-P011 Three differences
#11
★★★☆☆ Level 3 of 5
Let \(a+b+c=0\). Prove that \((a-b)^2+(b-c)^2+(c-a)^2=3(a^2+b^2+c^2)\).
Expand the left-hand side and replace \(ab+bc+ca\).
The left-hand side equals \(2(a^2+b^2+c^2-ab-bc-ca)\). Since \(a+b+c=0\), we have \(ab+bc+ca=-\frac{a^2+b^2+c^2}{2}\).
Substitution gives \(2\left(a^2+b^2+c^2+\frac{a^2+b^2+c^2}{2}\right)=3(a^2+b^2+c^2)\).
A good problem on transferring the basic identity to a new form.