  Working over a field of characteristic other than $2$, we examine a
relationship between quadrilaterals and the pencil of conics passing through
their vertices. Asymptotically, such a pencil of conics is what we call a
bisector field, a set ${\mathbb{B}}$ of paired lines such that each line $\ell$
in ${\mathbb{B}}$ simultaneously bisects each pair in ${\mathbb{B}}$ in the
sense that $\ell$ crosses the pairs of lines in ${\mathbb{B}}$ in pairs of
points that all share the same midpoint. We show that a quadrilateral induces a
geometry on the affine plane via an inner product, under which we examine
pencils of conics and pairs of bisectors of a quadrilateral. We show also how
bisectors give a new interpretation of some classically studied features of
quadrangles, such as the nine-point conic.
