Dear Mathlinkers, this is the Poncelet's porism. Sincerely Jean-Louis

Let $D$ be on the minor arc $BC$. Let $DE,DF$ be tangents to the incircle, with $E,F$ on the circumcircle and $E$ is on the arc $AC$ and $F$ is on the arc $AB$. Then it suffices to show $EF$ is tangent to the incircle. By Pascal's on hexagon $ABEDFC$, we get the intersection of $AB,DF$, call it $H$, the intersection of $ED,CA$, call it $G$ and the intersection of $BE,FC$, call it $J$, are collinear. In short, $H,J,G$ are collinear. Now use the converse of Brianchon on $FHBCGE$, since $FC,HG,BE$ are concurrent, and $FH,HB,BC,CG,GE$ are tangent to the incircle, we have $EF$ is tangent to the incircle, and we are done.