Hello DWH, and welcome to the Fourm:
I'm picturing your bay being made of two small windows on about a 45 degree angle; one on each side, with a larger window in the middle.
Another way they are sometimes built is with one wide window and two narrow windows squared on the end. I will answer for each type.
Type 1. I would cut the header at the widest point of the large window, extend a joist out on each side of that window, put single joist headers on each angle part and a double on the straight part; leaving the rest of the existing header in place. You would not have any part that makes the sub floor span more than 13" which would be a strong support.
Type 2. Again cut the header at each end and extend a joist out from each end, use a double header between the two and you will have sufficient strength to hold the addition. Also leave the existing header in place.
For either one, if the joist location you need does not land on an existing joist, you can go 4' to 6' into the existing floor space and add a block across between the existing joists then fasten the new joist to that block.
I turst that is clear enough but, if you have fruther questions, post back and we'll go at it a different way.