a, $\dfrac{a}{b}=\dfrac{c}{d}$
$⇔\dfrac{b}{a}=\dfrac{d}{c}$
$⇔\dfrac{b}{a}+1=\dfrac{d}{c}+1$
$⇔\dfrac{a+b}{a}=\dfrac{c+d}{c}$
$⇔\dfrac{a}{a+b}=\dfrac{c}{c+d}$
$⇒Đpcm$
b, $\dfrac{a}{b}=\dfrac{c}{d}$
$⇔\dfrac{b}{a}=\dfrac{d}{c}$
$⇔\dfrac{b}{a}-1=\dfrac{d}{c}-1$
$⇔\dfrac{b-a}{a}=\dfrac{d-c}{c}$
$⇔\dfrac{a}{b-a}=\dfrac{c}{d-c}$
$⇔\dfrac{a}{a-b}=\dfrac{c}{c-d}$
$⇒Đpcm$
c, Giả sử $\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}$
$⇔\dfrac{a}{a-b}+\dfrac{b}{a-b}=\dfrac{c}{c-d}+\dfrac{d}{c-d}$
$⇔\dfrac{b}{a-b}=\dfrac{d}{c-d}$
$⇔\dfrac{a-b}{b}=\dfrac{c-d}{d}$
$⇔\dfrac{a}{b}-1=\dfrac{c}{d}-1$
$⇔\dfrac{a}{b}=\dfrac{c}{d}$
$⇒$ Giả sử đúng
d, Từ $\dfrac{a}{b}=\dfrac{c}{d}$
$⇒\dfrac{a}{c}=\dfrac{b}{d}$
$⇒\dfrac{a^{2}}{c^{2}}=\dfrac{b^{2}}{d^{2}}=\dfrac{a^{2}-b^{2}}{c^{2}-d^{2}}$
Mà $\dfrac{a}{c}=\dfrac{b}{d}$
$⇒\dfrac{ab}{cd}=\dfrac{b^{2}}{d^{2}}$
Mà $\dfrac{b^{2}}{d^{2}}=\dfrac{a^{2}-b^{2}}{c^{2}-d^{2}}$
$⇒\dfrac{ab}{cd}=\dfrac{a^{2}-b^{2}}{c^{2}-d^{2}}$
