a) a2 + b2 + c2 = ab + bc + ac
\(\Rightarrow\) a2 + b2 + c2 - ab - bc - ac = 0
\(\Rightarrow\) 2(a2 + b2 + c2 - ab - bc - ac) = 0
\(\Rightarrow\) a2 + a2 + b2 + b2 + c2 + c2 - 2ab - 2bc - 2ac = 0
\(\Rightarrow\) (a2 - 2ab + b2) + (a2 - 2ac + c2) + (b2 - 2bc + c2) = 0
\(\Rightarrow\) (a - b)2 + (a - c)2 + (b - c)2 = 0
\(\Rightarrow\left\{{}\begin{matrix}\left(a-b\right)^2=0\\\left(a-c\right)^2=0\\\left(b-c\right)^2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}a-b=0\\a-c=0\\b-c=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}a=0\\a=c\\b=c\end{matrix}\right.\)
\(\Rightarrow\) a = b = c
b) (a + b + c)2 = 3(a2 + b2 + c2)
a2 + b2 + c2 + 2ab + 2bc + 2ac = 3a2 + 3b2 + 3c2
\(\Rightarrow\) 2ab + 2ac + 2bc = 2a2 + 2b2 + 2c2
\(\Rightarrow\) 0 = a2 + a2 + b2 + b2 + c2 + c2 - 2ab - 2bc - 2ac
Hay: a2 + a2 + b2 + b2 + c2 + c2 - 2ab - 2bc - 2ac = 0
\(\Rightarrow\)(a2 - 2ab + b2) + (a2 - 2ac + c2) + (b2 - 2bc + c2) = 0
\(\Rightarrow\) (a - b)2 + (a - c)2 + (b - c)2 = 0
\(\Rightarrow\left\{{}\begin{matrix}\left(a-b\right)^2=0\\\left(a-c\right)^2=0\\\left(b-c\right)^2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}a-b=0\\a-c=0\\b-c=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}a=0\\a=c\\b=c\end{matrix}\right.\)
\(\Rightarrow\) a = b = c
c) (a + b + c)2 = 3(ab + bc + ac)
a2 + b2 + c2 + 2ab + 2ac + 2bc = 3ab + 3bc + 3ac
\(\Rightarrow\) a2 + b2 + c2 = ab + ac + bc2
\(\Rightarrow\) 2(a2 + b2 + c2) = 2(ab + ac + bc)
\(\Rightarrow\) 2a2 + 2b2 + 2c2 = 2ab + 2bc + 2ac
\(\Rightarrow\) a2 + a2 + b2 + b2 + c2 + c2 - 2ab - 2bc - 2ac = 0
\(\Rightarrow\) (a2 - 2ab + b2) + (a2 - 2ac + c2) + (b2 - 2bc + c2) = 0
\(\Rightarrow\) (a - b)2 + (a - c)2 + (b - c)2 = 0
\(\Rightarrow\left\{{}\begin{matrix}\left(a-b\right)^2=0\\\left(a-c\right)^2=0\\\left(b-c\right)^2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}a-b=0\\a-c=0\\b-c=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}a=0\\a=c\\b=c\end{matrix}\right.\)
\(\Rightarrow\) a = b = c
CHÚC BN HOK TỐT(nhớ tik mik nha
)
a)Cmr : Nếu : a2 + b2 + c2 = ab + bc + ac thì a = b =c
Bài làm
2a2 + 2b2 + 2c2 = 2ab + 2bc + 2ca
=> 2a2 + 2b2 + 2c2 - 2ab - 2bc - 2ac = 0
=> ( a2 - 2ab + b2) + ( a2 - 2ac + c2) + ( b2 - 2bc + c2) =0
= > ( a - b)2 + ( a - c)2 + ( b -c)2 = 0
Vậy :
* ( a - b)2 = 0
* ( a - c)2 =0
* (b -c)2 =0
Suy ra :
* a =b
* a =c
* b = c
Suy ra : a = b =c ( đpcm)