$\frac{a}{b} = \frac{c}{d} $ if $ad = cb$, how to intuitively understand this? This works if you multiply both sides with $bd$ and cancel stuff out... But how does it work? 
When I look at it, I would never guess something like that is valid without resorting to the established arithmetic rules.
Maybe this is a nonsense question, should these things be analyzed in such a way or just accepted from the arithmetic rules? Since, that's the reason humanity has developed mathematics. To simplify and abstract things which would otherwise be out of the reach of our mind. 
Just a simple example, so, please... Set me straight. Thanks! 

Would a picture help? 
Note that $\frac{a}{b} = \frac{c}{d}$ by similar triangles.  The blue and green rectangle has area $ad$ while the green and yellow rectangle has area $bc$.  These are equal, namely fraction $\frac{a}{b} = \frac{c}{d}$ 
of the area of the big rectangle.
