I found this identity or sort of in my book, but there isn't a proof of this. The figure is attached below.

[![Sketch of the problem](https://i.stack.imgur.com/pFzm5.png)](https://i.stack.imgur.com/pFzm5.png)

As it can be seen it is a square where it has been traced four lines, going from all their corners to the midpoints of their opposing sides.

The letter $S$ represents I believe the surface. But I have no idea how one is three times of the other.

But again, I feel confused. How exactly the area of orange color is just one third of their contiguous trapezoid?.

Can someone explain this to me?. I tried to do it on my own and I believe the identity which comes into play is that when you have two triangles which share the same height and their bases are proportional to the other their respective areas also share the same proportion.

I appreciate that the answer can use some drawing so I can spot what's going on, because here are too many lines and I am confused. How can this be proven relying only in euclidean geometry?. Help, anyone?. Please.