**Bisector fields of quadrilaterals**, arXiv:2305.11762

**Authors: Bruce Olberding, Elaine A. Walker**

Abstract: 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 𝔹 of paired lines such that each line ℓ in 𝔹 simultaneously bisects each pair in 𝔹 in the sense that ℓ crosses the pairs of lines in 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. Submitted 19 May, 2023; originally announced May 2023.

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