# Problem on Concurrency of Lines and Ceva's Theorem

### Problem on Concurrency of Lines and Ceva's Theorem

Hi, I'm currently learning geometry on my own and just got stuck on a problem about concurrency of lines from Posamentier's Advanced Euclidean Geometry. So far the chapter has explained how to prove concurrency through Ceva's Theorem. This is the exercise:

In triangle ABC, AL, BM, and CN are concurrent at point P, where L, M, and N are points on BC, AC and AB respectively. Points S, Q, and R are midpoints of MN, ML, and NL, respectively. Prove that rays AS, BP, and CQ are concurrent.

For this exercise I tried to graph the problem and label further points on the triangles. However I didn't get any insights on how to prove concurrency from it as I think I might be missing some key property about the relationships between the triangles. It's clear to me that AM*CL*BN/CM*BL*AN = 1 and that SM=SN, NR=NL, ML=MQ, as well as the fact that triangles QRS and LMN are congruent and their paralell sides are on a 1:2 proportion. I have been trying for some hours now and would be thankful for any advice. This is what I tried:

https://www.geogebra.org/geometry/y5uu9gzb
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