Exercise 4, § 9

Let K be a field whose characteristic is different from 2 and 3, E an affine space over K, $a, b, c$ three points of E not on a straight line and $a', b', c'$ the respective midpoints of the ordered pairs $(b, c), (c, a)$ and $(a, b)$. Show that (in the notation of Exercise 3) the lines $D_{aa'}, D_{bb'},$ and $D_{cc'}$ pass through the barycentre of the three points $a, b, c$. What becomes of this property when K is of characteristic 2 or of characteristic 3? Generalize to a system of $n$ affinely independent points.*