Bicentric quadrilaterals through inversion (Q2871637)

If a quadrilateral admits a circumcircle (i.e., a circle that passes through the vertices), it is called cyclic; if it admits an incircle (i.e., a circle that touches the sides internally), it is called circumscriptible or tangential; if it admits a circumcircle and an incircle, it is called bicentric. Although cyclic quadrilaterals with several of their characterizations appear in Euclid's \textit{Elements}, the other two types did not appear until few centuries ago. Their main properties can be found in a series of papers by M. Josefsson that appear in Forum Geom. and in the references therein.NEWLINENEWLINEThe author of the paper under review gives new and smooth proofs of several notable properties of bicentric quadrilaterals using the powerful tool of inversion. Taking a bicentric quadriateral \(Q\), and letting \(R\) be the circumradius, \(r\) the inradius, and \(d\) the distance between the circumcenter and incenter, he establishes Fuss' formula \((R-d)^{-2}+(R+d)^{-2}=r^{-2}\), Carlitz' inequality \(\sqrt{2} r \leq R\), and a sharp form of Yun's generalization NEWLINE\[NEWLINE2\sqrt{2} r\leq \left(\sin \frac{A}{2} \cos \frac{B}{2} + \sin \frac{B}{2} \cos \frac{C}{2}+ \sin \frac{C}{2} \cos \frac{D}{2}+ \sin \frac{D}{2} \cos \frac{A}{2}\right)R \leq 2R.NEWLINE\]NEWLINE He also proves Poncelet's porism theorem stating that one can obtain a bicentric quadrilateral with the same circumcircle and incircle as \(Q\) by starting at any point on the circumcircle. He also establishes the orthogonality of the Newton line of \(Q\) with that of the quadrilateral whose vertices are the points where the incircle of \(Q\) touches its sides. Here, the Newton line is the line through the midpoints of the diagonals.