<PSA and <PTA are both right angles, and therefore supplementary. So <SPT and <TAS are supplementary and opposite angles in Quadrilateral PSAT are supplementary.
Now, <PST and <PAT have the same intercepted arc, which means the angles are congruent. Similary, points P, R, C, and S are cyclic.
So, <PCR and <PSR have the same intercepted arc, which means the angles are congruent.
By hiding some of the lines it is easier to see DPAB@DPCB [same intercepted arcs].
So, <PST congruent <PAT, <PCR congruent <PSR, and <PAB congruent <PCB.
Notice, <PAB is another name for <PAT and <PCB is another name for <PCR.
Using the transitive property we can establish <PST @ <PSR, which implies S, T, and R are collinear!