2023/05/19 by Olberding, Bruce, Walker, Elaine A.
#51A20 #51N10 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2305.11762
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 \mathbbB of paired lines such that each line ℓ in \mathbbB simultaneously bisects each pair in \mathbbB in the sense that ℓ crosses the pairs of lines in \mathbbB 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.