Suppose you were given
A= (x1,y1,z1), C = (x2,y2,z2)
Then O the centre of the face is at 0.5* (x1+x2,y1+y2,z1+z2)
And you must have D = ((x1+x2)*0.5+a,(y1+y2)*0.5+b,(z1+z2)*0.5)
and B = ((x1+x2)*0.5-a,(y1+y2)*0.5-b,(z1+z2)*0.5)
since you were told the Z coordinates were equal.
Now OA = ((x1-x2)*0.5,(y1-y2)*0.5),(z1-z2)*0.5)
while OD = (a,b,0)
But OD.OA = 0 so (x1-x2)*a+(y1-y2)*b = 0;
Also |OA|=|OD] so aa+bb = ((x1-x2)^2+(y1-y2)^2+(z1-z2)^2)*0.25
Eliminate either a or b using the equation coming from OD.OA = 0 and then
solve the quadratic to find a or b.