Let OX be a line through O, parallel to SA, meeting PA at X. Then OX is perpendicular to PA and it follows that X is the midpoint of PA.

Similarly, let TY be a line through T, parallel to SA, meeting AQ at Y. Then TY is perpendicular to PA and it follows that Y is the midpoint of AQ.

Now you should be able to see that, since S is the midpoint of OT, it follows that A is the midpoint of XY. Thus PX=XA=AY=YQ, and so PA=AQ.