From Gogeometry
Using :
- Define a = side of square ABCD
- Define h1,h2,h3 and h4 the heights respectively of S1, S2, S3 and S4
- h1+h3=a, h4-h2=a
- S=[ABCD]=a^2
- 2S1=a.h1, 2S3=a.h3
- 2S1+2S3=a(h1+h3)=a^2
- 2S2=a.h2, 2S4=a.h4
- 2S4-2S2=a(h4-h2)=a^2
- Therefore <b>S1+S3=S4-S2=S/2</b>