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@christianp The attached figure shows two pentagons inscribed in each other: each vertex of one pentagon lies on a line through a side of the other pentagon. You can't do this with quadrilaterals in the Euclidean plane, but you can in the complex projective plane: see https://en.wikipedia.org/wiki/M%C3%B6bius%E2%80%93Kantor_configuration

Happy birthday!