Competition problem — Geometry — Olympiad Maths Prep

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Geometry Difficulty 5.2 AIME, harder Prove it Belarus

The medians $AA_1$ and $BB_1$ of the triangle $ABC$ intersect at the point $G$ . Let $M$ and $N$ be the midpoints of the segments $GA$ and $GB$ , and let $K$ and $L$ be the midpoints of the segments $CB_1$ and $CA_1$ , respectively. The segments $KN$ and $LM$ intersect at the point $S$ .
Find the ratio $CS : SG$ .

Solution

Answer: $CS : SG = 2 : 1$ .

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