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collinear wanted, AR/RB =BP/PC=CQ/QA=CP_1/P_1B, centroid

Source: 2010 Indonesia MO Province P2 q1 OSP / Mexican Mathematical Olympiad 1997 OMM P2

December 10, 2020
geometryratioCentroidcollinear

Problem Statement

Given triangle ABCABC. Suppose PP and P1P_1 are points on BC,QBC, Q lies on CA,RCA, R lies on ABAB, such that ARRB=BPPC=CQQA=CP1P1B\frac{AR}{RB}=\frac{BP}{PC}=\frac{CQ}{QA}=\frac{CP_1}{P_1B} Let GG be the centroid of triangle ABCABC and K=AP1∩RQK = AP_1 \cap RQ. Prove that points P,GP,G, and KK are collinear.
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