Let $f( n )$ be the minimum over all convex planar n-gons of the number of different midpoints of the $\begin{pmatrix} n \\ 2 \end{pmatrix}$ line segments, or diagonals, between distinct vertices. It is proved that $f( n )$ is between approximately $0.8 \begin{pmatrix} n \\ 2 \end{pmatrix} $ and $0.9 \begin{pmatrix} n \\ 2 \\ \end{pmatrix} $. The upper bound uses the fact that the number of multiple midpoints, shared by two or more diagonals, can be as great as about $\begin{pmatrix} n \\ 2 \end{pmatrix}/10$. Cases for which the number of midpoints is at least $\lceil n( n - 2 )/2 \rceil + 1$, the number for a regular n-gon when n is even, are noted.