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Fair question.My immediate thought is yes, assuming you can see something moving at 99.9999% the speed of light (or its effects)
In addition to what they have already written to you, MomentaryLapseOfReason: Not only you would live for 5 seconds while the stationary others for 5 years; since you are running in a circle, the part of your body near the centre of rotation would live more time than the other part of your body... Please: don't ask me how it's possible.... 
Very cool LeeE but just for fun.Assume that you are moving at a straight velocity. Then all your bodyparts will have the same speed, right?So now you will have to make a course change. As a system, won't you still have the same speed overall as you make it?==you won't, would you?Would that make course changes impossible near 'c'?It would introduce the equivalent of 'tidal forces' on a body wouldn't it?And the inertia would differ too i presume.Always something new