Why $\frac{a}{b}=\frac{c}{d}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}\frac{a+c}{b+d}=\frac{a}{b}=\frac{c}{d}$?

In geometry class, this property was quickly introduced:

$\frac{a}{b}}={\displaystyle \frac{c}{d}}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{\displaystyle \frac{a+c}{b+d}}={\displaystyle \frac{a}{b}}={\displaystyle \frac{c}{d}$

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?

In geometry class, this property was quickly introduced:

$\frac{a}{b}}={\displaystyle \frac{c}{d}}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{\displaystyle \frac{a+c}{b+d}}={\displaystyle \frac{a}{b}}={\displaystyle \frac{c}{d}$

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?