31. The sides of a parallelogram are equal to $a$ and $b(a \neq b)$. Through the vertices of the obtuse angles of this parallelogram, lines perpendicular to the sides are drawn. These lines intersect to form a parallelogram similar to the original one. Find the cosine of the acute angle of the given parallelogram.
31. Consider two cases: 1 - when the bases of the perpendiculars lie on the sides of the parallelogram, and 2 - when one of the perpendiculars does not intersect the side to which it is dropped. In the 1 -st case, we arrive at a contradiction, and in the 2 -nd case, we get that $\cos \alpha=\frac{2 a b}{a^{2}+b^{2}}$.