<Tri-2024-2091>, hint

Let vectors of AB, AC be (AB), (AC).

AR/RB=BP/PC=CQ/QA=CP//P'B=t/1-t, 

(AP')=t(AB)+(1-t)(AC), Let AK/KQ=x/1-x, then (AK)=x(1-t)(AC)+(1-x)t(AB), (AK)=y(AP')

x=1/2/ AK=(1-t)1/2(AC)+t1/2(AB), AP=t(AC)+(1-t)(AB). (AG)=1/3(AB)+1/3(AC)

PG/GK=z/1-z.  (AG)=z(AK)+(1-z)(AC). Then z=2/3.

P,G,K are collinear.