How is it possible that all currency exchange rates are fixed w.r.t. to each other?
International currency markets are highly liquid, with near-instant transfer of knowledge between trading centres.

This means that any arbitrage opportunities tend to get resolved within seconds.

To put it simply, the thing you tried to do, of trading two currencies via a third, is something that lots of real-world currency traders, and their automatic trading programs, are trying to do all the time; so any discrepancies are quickly mopped up by the market.