I get 1/3. Without much of a formal math background, I'm not good at explanations, but . . . .
I labeled the SW corner of the square as A, NW as B, NE as C, SE as D, the midpoint of the north line as E, and the intersection as F.
ABE has an area of 1/4 and BCD has an area of 1/2, but they overlap within BEF.
Line AE increases its X value at twice the rate that BD decreases, so F is at y=2/3, giving BEF a height of 1/3 and a half-base of 1/4, and thus an area of 1/12.
1 - 1/4 - 1/2 + 1/12 = 1/3