SCHWEITZER 1964 Problem 4

Let $A_1,A_2,...,A_n$ be the vertices of a closed convex $n$ -gon $K$ numbered consecutively. Show that at least $n-3$ vertices $A_i$ have the property that the reflection of $A_i$ with respect to the midpoint of $A_{i-1}A_{i+1}$ is contained in $K$ . (Indices are meant $\textrm{mod} n\ .$ )