<Cir-2024-766>, hint

Since MS is perpendicular to lien p and MN is tangent to c' at R

O'PSR is a square of side length r'.

1> <NPS=<SMP, <RPS=<RTP ( MP cuts c' at T), <MRT=<RPT.

Sine <RTP=<MRT+<RMT, <RPN=<RPT=<RPM. PR bisects<MPN

2> IF PN bisects <RPS, then <NPS=<SMP=<MPO'. 

Hence 2r" cos<O'PM (90/4) sin <PMS(()/4)=PS=r'

r"sin45=r'