Its really useful if you need to fly somewhere, but there are no direct flights. You can look for intermediate stops close to the Great Circle route, and this will minimise your flying time. Of course you need to check on the schedules to minimise the holdover time.
For example, say I want to fly to Albuquerque from London Heathrow. This is the Great Circle route and is 4994 miles long:
There are no direct flights, but I can see from gc.kls2.com that my best best is going through Minneapolis St Paul to give a total airborne mileage of 4995 miles:
Now I could go via Washington Dulles, but thats 5304 miles ie about an extra half an our airborne:
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