$\begin{aligned} & {[\text { Median of a triangle }]} \\ & {[\quad \underline{\text { Trapezoid (other) }}]}\end{aligned}$

The larger base of the trapezoid is 24. Find its smaller base if the distance between the midpoints of the diagonals is 4.

## Answer

16. 

Prove that if the sides of a pentagon in order are 4, 6, 8, 7, and 9, then its sides cannot all be tangent to a single circle.
Let $x$ be one of the desired segments and express sequentially through $x$ the segments into which the points of tangency divide the sides of the pentagon.

## Solution

Assume that all sides of the given pentagon $A B C D E(A B=6, B C=8, C D=7, D E=9, E A=4)$ touch some circle. Denote the tangents from vertex $A$ as $x$. "Traverse" our pentagon, expressing sequentially the lengths of the tangents from vertices $B$ (equal to 6 - $x$), $C$ (equal to 8 - (6 - $x$) $=2$ + $x$), $D$ (equal to $7-(2+x)=5-x$), and $E$ (equal to $9-(5-x)=4+x$). We obtain that side $A E$ is divided by the point of tangency into segments $x$ and $4+x$, i.e., $x=0$, which is impossible.