$\\$
`b^2 = ac`
`-> b . b = a . c`
`-> b/a = c/b`
`-> a/b= b/c` (1)
`c^2 = bd`
`->c . c = b . d`
`-> c/b = d/c`
`-> b/c = c/d` (2)
Từ (1), (2)
`-> a/b=b/c = c/d`
Đặt `a/b=b/c=c/d=k (k \ne 0)`
`->` $\begin{cases} \dfrac{a}{b}=k\\\dfrac{b}{c}=k\\\dfrac{c}{d}=k\end{cases}$ `->` $\begin{cases} a=bk\\b=ck\\c=dk \end{cases}$
$\bullet$ `(a^3 + b^3 - c^3)/(b^3 +c^3 - d^3)`
`= ( (bk)^3 + (ck)^3 - (dk)^3)/(b^3 + c^3-d^3)`
`= (b^3k^3 + c^3k^3 - d^3k^3)/(b^3 +c^3-d^3)`
`= (k^3 (b^3 +c^3 - d^3) )/(b^3 + c^3-d^3)`
`= k^3` (1)
$\bullet$ `(a+b-c)^3/(b+c-d)^3`
`= (bk + ck - dk)^3/(b+c-d)^3`
`= [k (b+c-d)]^3/(b+c-d)^3`
`= [k^3 (b+c-d)^3]/(b+c-d)^3`
`= k^3` (2)
Từ (1), (2)
`-> (a^3 + b^3 - c^3)/(b^3 +c^3 - d^3) = (a+b-c)^3/(b+c-d)^3 (=k^3)`
`->` đpcm
