Because PR⊥AB is in R, PS⊥AC is in S, PR=PS, AP=PA.

So RPA≌ASP

So AS=AR

2. solution: from PR=PS, AP is the bisector of the angle CAB.

So angle CAP= angle BAP

In the triangle AQP, PQ=AQ, then the angle APQ=QAP). So the angle APQ= QAP= angle PAR, then AR//PQ (internal dislocation angles are equal and two straight lines are parallel).