Geometry without using trigonometry
So I'm supposed to find the area of the blue figure without using trigonometry. What do I have to do to find this?

Low bounty! Your deadline is also too short.

I think there's some missing info? Do the diagonal lines pass through the middle points of the edges of the square?
Answer
Look at the attached picture. We have that the triangles $ABC$ and $ADE$ are similar, so since $AE = 2AD$ then $AC = 2AB$. Now since there are so many right angles, all the segments marked with a single tick have the same length, and the ones marked with two ticks have twice that length. It follows that, if the big square has side lenght equal to $1$, then $AB = 1/6$. Now we can split the blue figure in four triangles, whose areas are easy to calculate: the green ones have area $1/9$, the pink ones $1/18$. Putting all those together, the area of the blue figure is \[ 2 \cdot \frac{1}{ 9} + 2 \cdot \frac{1}{18} = \frac{3}{9} = \frac{1}{3}. \]

I obviously meant BC instead of AB and DE instead of AD, but I guess you figured it out!
 answered
 316 views
 $5.89
Related Questions
 Find the volume of the solid obtained by rotating $y=x^2$ about yaxis, between $x=1$ and $x=2$, using the shell method.
 Deriving Ramanujan's Ellipse approximation
 Get area of rotated polygon knowing all coordinates and angle.
 Find the area bounded by the graphs of two functions
 Calculating aspect ratio limits of rotated rectangle within a rectangle
 Direction of forces acting on points A, B and C
 Geometry Problem about Hole Placement on PVC Pipe
 Three questions on the annihilator