Bulgarian Winter Tournament

Question

Points D and E are on sides BC and AC of ABC. Lines AD and BE intersect at point S. Point F is on side AB and lines FE and FD intersect line l passing through C and parallel to AB at points P and Q. Prove that if CP = CQ, then the points C, S and F lie on the same line.

Accepted Answer

From the similarities PEC FEA and CDQ BDF we obtain that CPAF = CEAE and CQBF = CDBD. Therefore CPCQ = CEAE AFBF BDDC. By condition CP = CQ, it follows that CEAE AFBF BDDC = 1. Applying Cheva's theorem for ABC and the points F, D, and E, it follows that the lines AD, BE, and CF intersect in one point, i.e. point S lies on CF.