(a - b)2 + (b - c)2 + (c - a)2 = 4(a2 + b2 + c2 - ab - ac - bc)
\(\Leftrightarrow\) a2 - 2ab + b2 + b2 - 2bc + c2 + c2 - 2ac + a2 = 4(a2 + b2 + c2 - ab - ac - bc)
\(\Leftrightarrow\) 2a2 + 2b2 + 2c2 - 2ab - 2ac - 2bc = 4(a2 + b2 + c2 - ab - ac - bc)
\(\Leftrightarrow\) 2(a2 + b2 + c2 - ab - ac - bc) = 4(a2 + b2 + c2 - ab - ac - bc)
\(\Leftrightarrow\) a2 + b2 + c2 - ab - ac - bc = 2(a2 + b2 + c2 - ab - ac - bc)
\(\Leftrightarrow\) 0 = 2(a2 + b2 + c2 - ab - ac - bc) - (a2 + b2 + c2 - ab - ac - bc)
\(\Leftrightarrow\) a2 + b2 + c2 - ab - ac - bc = 0
\(\Leftrightarrow\) 2(a2 + b2 + c2 - ab - ac - bc) = 0
\(\Leftrightarrow\) 2a2 + 2b2 + 2c2 - 2ab - 2ac - 2bc = 0
\(\Leftrightarrow\) (a2 - 2ab + b2) + (b2 - 2bc + c2) + (c2 - 2ac + a2) = 0
\(\Leftrightarrow\) (a - b)2 + (b - c)2 + (c - a)2 = 0
\(\Leftrightarrow\left\{{}\begin{matrix}\left(a-b\right)^2=0\\\left(b-c\right)^2=0\\\left(c-a\right)^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=b\\a=c\\c=a\end{matrix}\right.\)
\(\Rightarrow\)a = b = c
