Quadrilateral ABCD, going clockwise from upper left.
midpoint of AB is (X1+x2)/2, (y1+y2)/2
midpoint of CD is (x3+x4)/2, (y3+y4)/2
midpoint of AD is (x1+x3)/2. (y1+y3)/2
midpoint of BC is (x2+x3)/2, (y2+y3)/2
The midpoints of both bisectors are indeed (x1+x2+x3+x4)/2 , (y1+y2+y3+y4)/2
end of proof