Assuming you mean anti-meridian and not the equator then here goes (While all this works on a flattened map and should be good enough for your purpose, it's completely bung on a sphere. see note at the bottom).
What I've done in other cases is start at either point, and if the next point is more than 180 degrees to the east, I convert it so that it is less than 180 to the west like so
if(pointa.lon - pointb.lon > 180)
pointb.lon += 360:
else if (pointa.lon - pointb.lon < -180)
pointb.lon -= 360
At this time pointb.lon might be an invalid longitude like 190 but you can at least work out the mid-point between pointa and point b because they will be on a continuous scale, so you might have points 175 and 190. Then just get the mid-point between them as 182.5, then convert that to make sure it is within the usual limits and you get -177.5 as the latitude between the two points. Working out the latitude is easy.
Of course on a sphere this is wrong because the midpoint between (-180,89) and (180,89) is (0*,90) not (0,89).
* = could be anything