In any scalene triangle the three points of tangency of the incircle together with the three vertices can be used to define three new points which are, remarkably, always collinear. This line is called the Gergonne Line. Moreover cevians through these tangent points are always concurrent at a common point that, together with the incenter, defines a second line, the Soddy Line. Why should these lines be perpendicular? Such a geometric gem deserves a synthetic geometric proof. We use the classical theorems of Ceva and Menelaus to define these lines and then establish their perpendicularity by using a certain inversion.
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