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Israel 2012 Q6 - Equal segments

Source: Israel National Olympiad 2012 Q6

August 8, 2019
geometry

Problem Statement

Let A,B,C,OA,B,C,O be points in the plane such that angles ∠AOB,∠BOC,∠COA\angle AOB,\angle BOC, \angle COA are obtuse. On OA,OB,OCOA,OB,OC points X,Y,ZX,Y,Z respectively are chosen, such that OX=OY=OZOX=OY=OZ. On segments OX,OY,OZOX,OY,OZ points K,L,MK,L,M respectively are chosen.
The lines ALAL and BKBK intersect at point RR, which isn't on XYXY. The segment XYXY intersects AL,BKAL,BK at points R1,R2R_1,R_2.
The lines BMBM and CLCL intersect at point PP, which isn't on YZYZ. The segment YZYZ intersects BM,CLBM,CL at points P1,P2P_1,P_2.
The lines CKCK and AMAM intersect at point QQ, which isn't on ZXZX. The segment ZXZX intersects CK,AMCK,AM at points Q1,Q2Q_1,Q_2.
Suppose that PP1=PP2PP_1=PP_2 and QQ1=QQ2QQ_1=QQ_2. Prove that RR1=RR2RR_1=RR_2.
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