34. The segment connecting the midpoints of the diagonals of a trapezoid lies on its midline, so it is parallel to the bases. To prove that it is equal to half the difference of the bases, let's denote the lower base of the trapezoid by $a$, the upper base by $b$, and the desired segment by $x$. It is easy to see that the midline of the trapezoid will consist of segments $\frac{b}{2}, x, \frac{b}{2}$, but it is equal to $\frac{a+b}{2}$, therefore, $b+x=\frac{a+b}{2}$; from this we find $x=\frac{a-b}{2}$.