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arXiv · 2305.11762

Bisector fields of quadrilaterals

Bruce Olberding·Elaine A. Walker
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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 ${\mathbb{B}}$ 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.

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Bruce Olberding, Elaine A. Walker. 2023-05-19. Bisector fields of quadrilaterals. https://arxiv.org/abs/2305.11762

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