Tritangent centres, Pascal's theorem and Thébault's problem (Q1904264)

Let \(ABC\) be a triangle, with circumcircle \(\Sigma\). If circles with centers \(M\) and \(N\) are tangent to \(BC\), internally tangent to \(\Sigma\), and moreover share a common tangent through \(A\), then \(M\), \(N\), and the incenter of \(ABC\) are collinear. This beautiful result was conjectured in 1938 by V. Thébault, and was finally proved by K. B. Taylor in 1983. This paper, extending work of S.-C. Chou and R. Stark, contains several surprising and elegant results relating to Thébault's problem. The second section contains a simple but surprising ``Pascal-type'' theorem about octagons inscribed in conics. The third section contains a succesion of theorems, involving projective geometry and tritangent centers. These lead to a synthetic proof of an old result of Morley and Morley, showing the existence of a rectangular grid of eight lines linking the tritangent centers of the four triangles determined by a set of four points. Finally, these are brought together in a simple solution (independently discovered by R. Stark) to Thébault's problem. Other related results are also presented, including two interesting line configurations arising from the tritangent centers of the ten triangles formed by five points in general position on a circle.