Pairings of circles and Sawayama's theorem (Q2871646)

The Sawayama-Thebault theorem states that the centers \(O_1\) and \(O_2\) of two circles \(k_!\) and \(k_2\) defined by a cevian \(AW\) of a triangle \(ABC\) are collinear with the incenter \(I\) of \(ABC\). The circles \(k_1\) and \(k_2\) are by definition tangent to \(AW\), \(BC\) and the circumcircle of the triangle. In this article, the author studies some pairs of projectively related circles \((k_1,k_2)\) and reveal properties that in some particular cases, called Sawayama pairings, lead to Sawayama's theorem.