Show that $RU$ intersects $AB$ at the midpoint (of $AB$). $PA, PB$ are tangents. $PU$ is a reflection of $PS$ over $PO$ where $O$ is the center of the circle. Show that $RU$ intersects $AB$ at the midpoint (of $AB$).
It looks so obvious. And I can see a lot of similar triangles. But I can't really solve it.

Please avoid solutions that use inversive/projective geometry.

Note that $TS$ and $RU$ meets on $AB$ because $P$ is the pole of the line $AB$.
On the other hand $TS$ and $RU$ meets on the simmetry axis $PO$ because they corresponds under the reflection.
Thus $TS\cap RU=AB\cap PO$ and $AB\cap PO$ is the mid point of $AB$ because $OP$ is axis of $AB$.

