Question

CMR:ta có tỉ lệ thức a/b=c/d nếu có 1 trong các đẳng thức sau:

(a+b+c+d)(a-b-c+d)=(a-b+c-d)(a+b-c-d)

Answer 1

(a+b+c+d)(a-b-c+d)=(a-b+c-d)(a+b-c-d) => \(\frac{a+b+c+d}{a+b-\left(c+d\right)}=\frac{a-b+c-d}{a-b-\left(c-d\right)}\)

Áp dụng t/c của dãy tỉ số bằng nhau ta có:

\(\frac{a+b+c+d}{a+b-\left(c+d\right)}=\frac{a-b+c-d}{a-b-\left(c-d\right)}=\frac{\left(a+b+c+d\right)+\left(a-b+c-d\right)}{\left(a+b-\left(c+d\right)\right)+\left(a-b-\left(c-d\right)\right)}=\frac{\left(a+b+c+d\right)-\left(a-b+c-d\right)}{\left(a+b-\left(c+d\right)\right)-\left(a-b-\left(c-d\right)\right)}\)

=> \(\frac{a+c}{a-c}=\frac{b+d}{b-d}\)=> \(\frac{a+c}{b+d}=\frac{a-c}{b-d}\) => \(\frac{\left(a+c\right)+\left(a-c\right)}{\left(b+d\right)+\left(b-d\right)}=\frac{\left(a+c\right)-\left(a-c\right)}{\left(b+d\right)-\left(b-d\right)}\)=> \(\frac{a}{b}=\frac{c}{d}\)

Vậy...