Bisector fields of quadrilaterals

A bisector field is a maximal set 𝔹 of paired lines in the plane such that each line in each pair crosses the other pairs in 𝔹 in pairs of points that all share the same midpoint. We showed in a previous article that bisector fields are precisely the sets of pairs of lines that occur as asymptotes of hyperbolas from a pencil of affine conics, along with pairs of parallel lines arising from degenerate parabolas in the pencil. In this article we give a different application, this time to complete quadrilaterals and their nine-point conics. We show that every complete quadrilateral generates a bisector field as the set of bisectors of the quadrilateral paired according to an orthogonality condition in a geometry determined by the quadrilateral. The nine-point conic, so named because it passes through nine distinguished points of the quadrilateral, is shown to be the locus of midpoints of the bisector field associated to the quadrilateral, thus giving an interpretation of the other points on the nine-point conic. Our approach is analytic, and our results hold over any field of characteristic other than 2.