Edit: Found a newer article  which does point a working regex101 link 
Oh, I thought we were past that already...
a + b = c, where a = b = c
* a = b = c = 0
* a = b = c = k/inf (where k < inf and k > -inf)
Off-topic but are there other solutions? I checked in Wolfram Alpha and nothing...
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.