Elegant 12

When we divide one of the four blue regions into smaller right triangles, we can see that all of these triangles are congruent as they are right-angled and have equal non-hypotenuse sides. Thus, 6 triangles will make up one blue corner region, and the total blue area will equal to 24 triangles.

Now considering the triangles touching the red tangents, we can see that at each corner, any of the three triangle is also congruent to the blue triangle. Therefore, the total area outside the tangents = (4\times (6+3)) = 36 triangles.

Then if draw perpendicular lines joining the square midpoints, we will see that red tangents act as the diagonals of the 4 congruent squares. Therefore, the area of the big square is halved by the red tangents, and the total area of the big square = 72 triangles.

Hence, the fraction of the blue area = 24/72 = 4/12 or 1/3.

Note: Each "reduced" corner area is 1/12 of the big square. It's an elegant 12 indeed!

When we notice the small squares and isosceles right triangles (the marked segments are the same according to assumptions), it is easy to see that the big square is composed of $6 \times 6$ small squares. It means that the ratio of the blue area to the area of the big square is: $\frac{4 \cdot 3}{6 \cdot 6} = \frac{4}{12}$ .