Question

Cho a,b,c,d là các số khác 0 và (a+b+c+d)(a-b-c+d)=(a-b+c-d)(a+b-c-d).CMR:a/c=b/d

Answer 1

=> \(\dfrac{a+b+c+d}{a-b+c-d}=\dfrac{a+b-c-d}{a-b-c+d}\)

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

=> \(\dfrac{2a+2b}{2a-2b}=\dfrac{2c+2d}{2c-2d}\)

=>\(\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}\)

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

=> ac-ad+bc-bd = ac+ac-bc-bd

=>bc-ad = ad-bc

=> bc-ad-ad+bc = 0

=>2(bc-ad) = 0

=> bc- ad = 0

=> bc = ad

=>\(\dfrac{a}{c}=\dfrac{b}{d}\)