Let $(C)$ be a circle , and $ABCD$ be a quadrilateral inscribed in $(C)$
Let $P$ be the intersection of $(AD)$ and $(BC)$. And $Q$ the intersection of $(AB)$ and $(CD)$
Let $S$ and $T$ be points in $(C)$ such that $(PS)$ and $(PT)$ are tangents to $(C)$
this problem can be done by Lahire theorem or projective geometry or polarisation.
But I wanna if there is a simple solution by angle chasing or radical axis
here is what I think we should do :
$$\widehat{OSP}=180-\widehat{OTP}=90$$ so OSPT is cyclic
so the problem reduces itself to prove that Q lies on the radical axis of $(C)$ and the circle where $OSPT$ is inscribed.
N.B: This problem is taken from a preparation test for IMO 2020 in Morocco.