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Working paper

Dual quadrangles in the plane

arxiv.org. math. Cornell University, 2019. No. 1911.09321.
We consider quadrangles of perimeter 2 in the plane with marked directed edge. To such quadrangle $Q$ a two-dimensional plane $\Pi\in\mathbb{R}^4$ with orthonormal base is corresponded. Orthogonal plane $\Pi^bot$ defines a plane quadrangle $Q^\circ$ of perimeter 2 and with marked directed edge. This quadrangle is defined uniquely (up to rotation and symmetry). Quadrangles $Q$ and $Q^\circ$ will be called dual to each other. The following properties of duality are proved: a) duality preserves convexity, non convexity and self-intersection; b) duality preserves the length of diagonals; c) the sum of lengths of corresponding edges in $Q$ and $Q^\circ$ is 1.