7 regions

place all the digits 1 to 7, one in each region
so that the three circles all have the same total

I'm fairly confident that there are 18 solutions altogether and these are not too hard to find:

• 1 way totalling 13
• 3 ways totalling 14
• 2 ways totalling 15
• 6 ways totalling 16
• 2 ways totalling 17 
• 3 ways totalling 18 
• 1 way totalling 19

why is there a symmetry to these numbers of possibilities?

proofs:

• 1 must go in the centre for totals of 13 
• 7 must go in the centre for totals of 19

impossibility proofs:

• you cannot make totals greater than 19 
• you cannot make totals less than 13
• circles cannot total 16 with a 4 in the centre

place letters (a to g) in the regions, going left to right and then down
prove that:

• a + b = g + f
• b + c = d + g
• c + f = a + d
• b + e + g (etc) must be even