1. Ta có:
\(\left(a^2+b^2\right)\left(x^2+y^2\right)=\left(ax+by\right)^2\)
=> \(a^2x^2+a^2y^2+b^2x^2+b^2y^2=a^2x^2+2axby+b^2y^2\)
=> \(a^2y^2+b^2x^2=2axby\)
=> \(a^2y^2+b^2x^2-2axby=0\)
=> \(a^2y^2+b^2x^2-2aybx=0\)
=> \(\left(ay-bx\right)^2=0\)
Mà \(\left(ay-bx\right)^2\ge0\)
Dấu '' = '' xảy ra \(\Leftrightarrow\) \(ay-bx=0\)
\(\Leftrightarrow\) \(ay=bx\)
\(\Leftrightarrow\) \(\dfrac{a}{x}=\dfrac{b}{y}\)
2. Ta có:
\(a^2+b^2+c^2=ab+bc+ac\)
=> \(2a^2+2b^2+2c^2=2ab+2bc+2ac\)
=> \(2a^2+2b^2+2c^2-2ab-2bc-2ac=0\)
=> \(\left(a^2-2ab+b^2\right)+\left(a^2-2ac+c^2\right)+\left(b^2-2bc+c^2\right)=0\)
=> \(\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2=0\)
Ta thấy:
\(\left(a-b\right)^2\ge0\); \(\left(a-c\right)^2\ge0\); \(\left(b-c\right)^2\ge0\)
=> \(\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2\ge0\)
Mà \(\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2=0\)
Dấu '' = '' xảy ra \(\Leftrightarrow\) \(\left\{{}\begin{matrix}a-b=0\\a-c=0\\b-c=0\end{matrix}\right.\)
\(\Leftrightarrow\)\(\left\{{}\begin{matrix}a=b\\a=c\\b=c\end{matrix}\right.\)
\(\Leftrightarrow\) a = b = c
