We will be bringing together ideas from classical projective geometry, triangle geometry, graph theory and hypergroup theory in order to understand a remarkable generalization of the Schiffler point in 20th century triangle geometry. The Schiffler point is the point of concurrence of the Euler lines of the three triangles formed by two vertices of a triangle and the Incenter. In 2003 L. Emelyanov and T. Emelyanov discovered another pleasant description of the Schiffler point involving the Circumcentre, which we look to generalizing, leading to a surprising five-fold symmetry.
We will see that dihedral orderings of five objects leads to a graph which we call the twisted icosahedron, which controls the tangential meetings of 12 beautiful conics. These conics are associated to five general points in the plane and their 15 associated generalized Schiffler points as meets of the tangents to the unique conic passing through those 5 points.
Finally we will compare the hypergroup structures of the twisted icosahedron and the regular icosahedron. This talk describes joint work with Nguyen Le, and a notable feature of the theory is that it is purely algebraic and works in fact over a general field and with an arbitrary non-degenerate quadratic form.