If you break a centre drill in a piece that you have in the lathe, you can recover the piece by making the item above.
This only works if you are going to drill a hole bigger than the hole that the above tool leaves (it depends on the job
really as the hole this tool leaves isn't that deep). OK, grind down the broken centre drill till it looks like the pictures above. Make sure that the cutting tip is offset enough to miss the broken tip in the job.
Stick it in your chuck. Start the lathe and VERY carefully press the tool against the job.
It should start to cut a ring around the broken drill tip. Soon you wil be deep enough to knock the broken part out.

This has saved my skin a few times.