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Đặt \(\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}=k\) thì \(x=ak,y=bk,z=ck\)
\(\dfrac{bz-cy}{a}=\dfrac{bck-bck}{a}=0\) __( 1 )__
\(\dfrac{cx-az}{b}=\dfrac{ack-ack}{b}=0\) __( 2 )__
\(\dfrac{ay-bx}{c}=\dfrac{abk-abk}{c}=0\) __( 3 )__
Từ ( 1 ), ( 2 ), ( 3 ) suy ra \(\dfrac{bz-cy}{a}=\dfrac{cx-az}{b}=\dfrac{ay-bx}{c}\)
Đặt \(t=\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}\Rightarrow x=at,y=bt,z=ct\)
\(\dfrac{bz-cy}{a}=\dfrac{bct-bct}{a}=0\), tương tự ta có: \(\dfrac{cx-az}{b}=\dfrac{ay-bx}{c}=0\)
Do đó \(\dfrac{bz-cy}{a}=\dfrac{cx-az}{b}=\dfrac{ay-bx}{c}\)
Ta có :
\(\dfrac{cy-bx}{x}=\dfrac{az-cx}{y}=\dfrac{bx-ay}{z}=\dfrac{bxz-cxy+cxy-ayz+ayz-bxz}{ax+by+cz}=0\)
\(\Rightarrow\dfrac{cy-bz}{x}=0\) \(\Rightarrow cy=bz\) \(\Rightarrow\) \(\dfrac{b}{y}=\dfrac{c}{z}\left(1\right)\)
\(\Rightarrow\dfrac{az-cx}{y}=0\) \(\Rightarrow az=cx\) \(\Rightarrow\dfrac{a}{x}=\dfrac{c}{z}\left(2\right)\)
Từ (1) và (2) suy ra : \(\dfrac{a}{x}=\dfrac{b}{y}=\dfrac{c}{z}\)
Ta có :
\(\dfrac{cy-bz}{x}=\dfrac{az-cx}{y}=\dfrac{bx-ay}{z}=\dfrac{bxz-cxy+cxy-ayz+ayz-bxz}{ax+by+cz}=0\)
\(\Rightarrow\dfrac{cy-bz}{x}=0\Rightarrow cy=bz\Rightarrow\dfrac{b}{y}=\dfrac{c}{z}\left(1\right)\)
\(\Rightarrow\dfrac{az-cx}{y}=0\Rightarrow az=cx\Rightarrow\dfrac{a}{x}=\dfrac{c}{z}\left(2\right)\)
Từ (1) và (2) suy ra:\(\dfrac{a}{x}=\dfrac{b}{y}=\dfrac{c}{z}\)
ta có: x/a = y/b =z/c =xa/a^2 =yb/b^2 =zc/c^2 = (ax+by+cz)/(a^2+b^2+c^2)
=>x/a = (ax+by+cz)/(a^2+b^2+c^2) (1)
mặt khác ta có: x/a=y/b=z/c <=> x^2/a^2 =y^2/b^2 =z^2/c^2 = (x^2+y^2+z^2 ) / (a^2+b^2+c^2)
=>x^2/a^2 = (x^2+y^2+z^2 ) / (a^2+b^2+c^2) (2)
từ (1) và (2) ta => (ax+by+cz)^2/(a^2+b^2+c^2)^2 = (x^2+y^2+z^2 ) / (a^2+b^2+c^2)
=> (x^2+y^2+z^2).(a^2+b^2+c^2)=(ax+by+cz)^2 => đpcm
\(\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}=k\Rightarrow x=ak,y=bk,z=ck\)
\(\dfrac{bz-cy}{a}=\dfrac{b.ck-c.bk}{a}=\dfrac{0}{a}=0\)(1)
\(\dfrac{cx-az}{b}=\dfrac{c.ak-a.ck}{b}=\dfrac{0}{b}=0\)(2)
\(\dfrac{ay-bz}{c}=\dfrac{a.bk-b.ak}{c}=\dfrac{0}{c}=0\)(3)
từ (1),(2) và(3) suy ra \(\dfrac{bz-cy}{a}=\dfrac{cx-az}{b}=\dfrac{ay-bx}{c}\left(đpcm\right)\)