\(a^4+b^4+c^4=2a^2b^2+2a^2c^2+2b^2c^2\)
\(\Leftrightarrow a^4+b^4+c^4-2a^2b^2-2a^2c^2-2b^2c^2=0\)
\(\Leftrightarrow\left(a^4-2a^2b^2+b^4\right)+\left(b^4-2b^2c^2+c^4\right)+\left(c^4-2c^2a^2+a^4\right)-a^4-b^4-c^4=0\)
\(\Leftrightarrow\left(a^2-b^2\right)^2+\left(c^2-b^2\right)^2+\left(c^2-a^2\right)^2-a^4-b^4-c^4=0\)
\(\Leftrightarrow\left(a-b\right)^2c^2+a^2\left(b+c\right)^2+b^2\left(c+a\right)^2-a^4-b^4-c^4=0\)
\(\Leftrightarrow c^2\left[\left(a-b\right)^2-\left(a+b\right)^2\right]+a^2\left[\left(b+c\right)^2-a^2\right]+b^2\left[\left(c+a\right)^2-b^2\right]=0\)
\(\Leftrightarrow c^2\left[\left(a-b\right)^2-\left(a+b\right)^2\right]+a^2\left[\left(b+c\right)^2-\left(c-b\right)^2\right]+b^2\left[\left(c+a\right)^2-\left(c-a\right)^2\right]=0\)
\(\Leftrightarrow-4abc^2+4a^2bc+4ab^2c=0\)
\(\Leftrightarrow4abc\left(a+b-c\right)=0\)
\(\Leftrightarrow0=0\)(luôn đúng)
=>đpcm
