IL-GILLIS 2012 Problem 6

Let $A,B,C,O$ be points in the plane such that angles $\angle AOB,\angle BOC, \angle COA$ are obtuse. On $OA,OB,OC$ points $X,Y,Z$ respectively are chosen, such that $OX=OY=OZ$ . On segments $OX,OY,OZ$ points $K,L,M$ respectively are chosen.

The lines $AL$ and $BK$ intersect at point $R$ , which isn't on $XY$ . The segment $XY$ intersects $AL,BK$ at points $R_1,R_2$ .

The lines $BM$ and $CL$ intersect at point $P$ , which isn't on $YZ$ . The segment $YZ$ intersects $BM,CL$ at points $P_1,P_2$ .

The lines $CK$ and $AM$ intersect at point $Q$ , which isn't on $ZX$ . The segment $ZX$ intersects $CK,AM$ at points $Q_1,Q_2$ .

Suppose that $PP_1=PP_2$ and $QQ_1=QQ_2$ . Prove that $RR_1=RR_2$ .