>> Let $ABCD$ be a quadrilateral, $P$ be a point in the plane let $E$, $F$ be the projections of the incenters of  triangles $\triangle CPB$, $\triangle BPA$ onto $PB$ respectively; Let $G$, $H$ be the projections of the incenters of triangles $\triangle APD$, $\triangle DPC$ onto $PD$ respectively. *Then $ABCD$ is a tangential quadrilateral if only if with **every** arbitrary point P in the plane $EF=GH$.*



Following picture: $ABCD$ be a tangential quadrilateral if only if with $P$ be **every** arbitrary point in the plane then two red segment lengths are equal

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/ntY8K.png