Answer's a little more complicated than regular formulas, but what do I need to do?

If you can live with circular references until the values are entered:

Cell A2:
=B2+C2

Cell B2:
=A2+C2

Cell C2:
=A2-B2

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Otherwise separate the input area (A2:C100) and the output area (D2:D100)

Cell D2:
=IF(COUNTA(A2:C2)<2,"",IF(A2="",B2+C2,A2))

Cell E2:
=IF(COUNTA(A2:C2)<2,"",IF(B2="",A2+C2,B2))

Cell F2:
=IF(COUNTA(A2:C2)<2,"",IF(C2="",A2-B2,C2))

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Alternative: A macro