Not sure where you get those weird infinity results from, but the "a = b = c = 0" solution is easily show:
Note that "a + b = c" implies "b = c - a", and given our assumption that "a = b = c", we can rewrite as "a = a - a = 0". This obviously results in "a = b = c = 0".
EDIT: Modifying the proof above, it becomes apparent that this is a property of groups (most number systems are groups): Again "a + b = c", iff "a + a = a" iff "a + a + (-a) = a + (-a)" which is "a = e".
Note that addition over IEEE754 floats do not constitute a group, (in part) because "inf + inf == inf" evaluates to true.
Note that "a + b = c" implies "b = c - a", and given our assumption that "a = b = c", we can rewrite as "a = a - a = 0". This obviously results in "a = b = c = 0".
EDIT: Modifying the proof above, it becomes apparent that this is a property of groups (most number systems are groups): Again "a + b = c", iff "a + a = a" iff "a + a + (-a) = a + (-a)" which is "a = e".
Note that addition over IEEE754 floats do not constitute a group, (in part) because "inf + inf == inf" evaluates to true.