<s>[INST] Why $\frac {a}{b} = \frac {c}{d} \implies \frac {a+c}{b+d} = \frac {a}{b} = \frac {c}{d}$? In geometry class, this property was quickly introduced:

$$\dfrac {a}{b} = \dfrac {c}{d} \implies \dfrac {a+c}{b+d} = \dfrac {a}{b} = \dfrac {c}{d} $$

It usually avoids some boring quadratic equations, so it's useful but I don't really get it. Seems trivial but I can't prove. 
How to demonstrate it?
 [/INST] $$\frac{a}{b}=\frac{c}{d}\Rightarrow\frac{a+c}{b+d}=\frac{a+\frac{ad}{b}}{b+d}=\frac{ab+ad}{b(b+d)}=\frac{a}{b}=\frac{c}{d}$$
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