Exercise 3, § 9

Let K be a commutative field of characteristic $\neq 2$, E an affine plane over K and $a, b, c, d$ four points of E any three of which do not lie on a straight line. Let $D_{xy}$ denote the line passing through two distinct points $x, y$ of E. Suppose that the lines $D_{ab}$ and $D_{cd}$ have a common point $e$ and that $D_{ad}$ and $D_{bc}$ have a common point $f$. Show that the midpoints of the three ordered pairs $(a, c), (b, d), (e, f)$ lie on a straight line. What becomes of this property when $D_{ab}$ and $D_{cd}$ are parallel or when $D_{ad}$ and $D_{bc}$ are parallel? Consider the case where K is a field with 3 elements.*