Competition problem — Geometry — Maths Olympiad Prep

Library / /32 of 214

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: $CS : SG = 2 : 1$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.