More Olympic Ring Addition

Given the five olympic rings how can the digits one through nine be placed within the nine regions (five non-overlapping ring regions and four overlapping regions shared between two rings) so that each ring contains a successively incrementing total?

Or alternatively, using the labels A through I for the regions, how can the numbers one through nine be assigned to the variables such that:

A+B = B+C+D-1 = D+E+F-2 = F+G+H-3 = H+I-4

How many different totals can fit in the first circle?  Find a complete solution for each.

Source: Original. Similar to previous POTW July 25, 1997

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