Thébault's theorem (Q1025797)

Thébault's Theorem states that the centers of two circles tangent to the circumcircle, a Cevian \(AT\) and the side \(BC\) of a triangle \(ABC\) span a line which contains the incenter of \(ABC\). Note that a rigorous formulation needs to take into account the existence of four circles tangent to a given circle and two lines. Thébault's result only holds for two of them. The theorem has been published as a problem in [\textit{V.~Thébault}, Am. Math. Mon. 45, 482--483 (1938)]. Several proofs of different flavor are known. The authors present an analytic proof which is based on a characterization of circles tangent to the side of a triangle and its circumcircle. They derive an elementary construction of Thébault's circles and discuss several straightforward consequences. Finally, the authors ask for a spatial generalization of Thébault's Theorem. Two obvious ideas are falsified so that the question remains open.