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Giải:

Đặt : \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}=k\)  => \(\hept{\begin{cases}x=ak\\y=bk\\z=ck\end{cases}}\)

Khi đó, ta có:

\(\frac{b.ck-c.bk}{a}=\frac{0}{a}=0\) (1)

\(\frac{c.ak-a.ck}{b}=\frac{0}{b}=0\) (2)

\(\frac{a.bk-b.ak}{c}=\frac{0}{c}=0\) (3)

Từ (1); (2); (3) suy ra \(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}\)

Ta có : \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\)

\(\Rightarrow\hept{\begin{cases}bz=cy\\cx=az\\ay=bx\end{cases}}\)

\(\Rightarrow bz-cy=cx-az=ay-bx=0\)

\(\Rightarrow\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}=0\left(ĐPCM\right)\)

Ta có \(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}\)

\(\Rightarrow\frac{\left(bz-cy\right).x}{ax}=\frac{\left(cx-az\right)y}{by}=\frac{\left(ay-bx\right).z}{cz}\)

\(\Rightarrow\frac{bxz-cxy}{ax}=\frac{cxy-azy}{by}=\frac{ayz-bxz}{cz}\)

Áp dụng tính chất của dãy tỉ số bằng nhau ta có:

\(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}=\frac{bxz-cxy+cxy-ayz+ayz-bxz}{ax+by+cz}\)

Suy ra:

        bz - cy = 0                        (1)

        cx - az = 0                        (2)

        ay - bx = 0                        (3)

Từ (1) ta có: \(bz=cy\Rightarrow\frac{y}{b}=\frac{z}{c}\left(I\right)\)

Từ (2) ta có: \(cx=az=\frac{z}{c}=\frac{x}{a}\left(II\right)\)

Từ (3) ta có: \(ay=bx=\frac{x}{a}=\frac{y}{b}\left(III\right)\)

Từ (I), (II), (III) => x: y: z = a: b: c

\(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}\)

\(\Rightarrow\frac{abz-acy}{a^2}=\frac{bcx-abz}{b^2}=\frac{acy-bcx}{c^2}\)

Áp dụng tính chất dãy tỉ số bằng nhau , ta có :

\(\frac{abz-acy}{a^2}=\frac{bcx-abz}{b^2}=\frac{acy-bcx}{c^2}=\frac{abz-acy+bcx-abz+acy-bcx}{a^2+b^2+c^2}=\frac{0}{a^2+b^2+c^2}=0\)

\(\Rightarrow\hept{\begin{cases}bz-cy=0\\cx-az=0\\ay-bx=0\end{cases}}\Rightarrow\hept{\begin{cases}bz=cy\\cx=az\\ay=bx\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}\frac{y}{b}=\frac{z}{c}\\\frac{x}{a}=\frac{z}{c}\\\frac{y}{b}=\frac{x}{a}\end{cases}}\Rightarrow\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\)

* C1 :(bz - cy)/a = (abz - acy)/a²

(cx - az)/b = (bcx - abz)/b²

(ay - bx)/c = (acy - bcx)/c²

Mà (bz - cy)/a = (cx - az)/b = (ay - bx)/c

=>(abz - acy)/a² = (bcx - abz)/b² = (acy - bcx)/c² = (abz - acy + bcx - abz + acy - bcx)/a² + b² + c² = 0

=>(bz - cy)/a = (cx - az)/b = (ay - bx)/c = 0

=>bz - cy = cx - az = ay - bx = 0

*Xét bz - cy = 0

=>bz = cy

=>z/c = y/b

Chứng minh tương tự = >x/a = y/b ; x/a = z/c

=> x/a = y/b = z/c

*C2 : 

(bz - cy)/a = (abz - acy)/ax

(cx - az)/by = (bcx - abz)/by

(ay - bx)/cz = (acy - bcx)/cz

Làm tương tự như C1

\(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}\)

Suy ra: \(\frac{bxz-cxy}{ax}=\frac{cxy-azy}{bx}=\frac{azy-bxz}{cx}\). Áp dụng tính chất dãy tỉ số bằng nhau có: 

\(\frac{bxz-cxy}{ax}=\frac{cxy-azy}{bx}=\frac{azy-bxz}{cx}=\frac{\left(bxz-cxy\right)+\left(cxy-azy\right)+\left(azy-bxz\right)}{ax+bx+cx}\)

\(=\frac{\left(bxz-bxz\right)-\left(cxy-cxy\right)-\left(azy-azy\right)}{ax+by+cz}=\frac{0}{ax+by+cz}\)

Suy ra: \(\hept{\begin{cases}bz-cy=0\\cx-az=0\\ay-bx=0\end{cases}\Leftrightarrow}\hept{\begin{cases}bz=cy\\cx=az\\ay=bx\end{cases}}\)

Áp dụng tính chất tỉ lệ thức ta được: \(\hept{\begin{cases}\frac{y}{b}=\frac{z}{c}\\\frac{z}{c}=\frac{x}{a}\\\frac{x}{a}=\frac{y}{b}\end{cases}\Leftrightarrow\frac{x}{a}=\frac{y}{b}=\frac{z}{c}^{\left(đpcm\right)}}\)

vi bz-cy/a=cx-az/b=ay-bx/c=>a(bz-cy)/a^2=b(cx-az)/b^2=c(ay-bx)/c^2

=>abz-acy/a^2=bcx-abz/b^2=cay-cbx/c^2=>abz-acy+bcx-abz+cay-cbx/a^2+b^2+c^2

=>o/a^2+b^2+c^2=0

=>bz-cy=0=>y/b=z/c(1)

cx-az=o=>x/a=z/c(2)

từ (1) và (2) =>x/a=y/b=z/c

mk k viết đề nha bạn!

\(=>\frac{a\left(bz-cy\right)}{a^2}=\frac{b\left(cx-az\right)}{b^2}=\frac{c.\left(by-ax\right)}{c^2}\)

\(=>\frac{abz-acy}{a^2}=\frac{bcx-abz}{b^2}=\frac{cay-bcx}{c^2}\)\(=\frac{abz-acy+bcx-acz+cay-bcx}{a^2+b^2+c^2}=0\)

\(=>\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bc}{c}=0\)

=> bz - cy = cx - az = ay - bx = 0

+) bz - cy = 0 => bz = cy => y / b = z/c 

+) cx - az = 0 => cx = az => x / a = z/ c
=> x / a = y / b = z/ c ( dpcm )

\(\dfrac{bz-cy}{a}=\dfrac{cx-az}{b}=\dfrac{ay-bx}{c}\)

\(=\dfrac{a\left(bz-cy\right)}{a.a}=\dfrac{b\left(cx-az\right)}{b.b}=\dfrac{c\left(ay-bx\right)}{c.c}\)

\(=\dfrac{abz-acy}{a^2}=\dfrac{bcx-baz}{b^2}=\dfrac{cay-cbx}{c^2}\)

\(=\dfrac{abz-acy+bcx-baz+cay-cbx}{a^2+b^2+c^2}\)

\(=\dfrac{0}{a^2+b^2+c^2}\)

\(=0\)

\(\Rightarrow\left\{{}\begin{matrix}\dfrac{bz-cy}{a}=0\Rightarrow bz-cy=0\Rightarrow bz=cy\Rightarrow\dfrac{b}{y}=\dfrac{c}{z}\\\dfrac{cx-az}{b}=0\Rightarrow cx-az=0\Rightarrow cx=az\Rightarrow\dfrac{c}{z}=\dfrac{a}{x}\\\dfrac{ay-bx}{c}=0\Rightarrow ay-bx=0\Rightarrow ay=bx\Rightarrow\dfrac{a}{x}=\dfrac{b}{y}\end{matrix}\right.\)

\(\Rightarrow\dfrac{a}{x}=\dfrac{b}{y}=\dfrac{c}{z}\)

\(\Rightarrow a:b:c=x:y:z\)

Đặt \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}=k\)

\(\Rightarrow\hept{\begin{cases}x=ak\\y=bk\\z=ck\end{cases}}\)

Ta có \(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}\)

\(\Rightarrow\frac{bck-cbk}{a}=\frac{cak-ack}{b}=\frac{abk-bak}{c}\)

\(\Rightarrow\frac{0}{a}=\frac{0}{b}=\frac{0}{c}\)

\(\Rightarrow0=0=0\)(đpcm)

 \(\frac{x}{a}=\frac{y}{b}=\frac{z}{c}\Rightarrow\hept{\begin{cases}bx=ay\\cx=az\\cy=bz\end{cases}\Rightarrow\hept{\begin{cases}ay-bx=0\\cx-az=0\\bz-cy=0\end{cases}\Rightarrow}}\)\(\frac{bz-cy}{a}=\frac{cx-az}{b}=\frac{ay-bx}{c}=0\left(đpcm\right)\)