In a triangle ABC, points D and E are on segments BC and AC such that BD = 3DC and AE = 4EC. Point P is on line ED such that D is the midpoint of segment EP. Lines AP and BC intersect at point S. Find the ratio BS/SD (RMO 2013, Mumbai Region)
Let (F) denote the midpoint of the segment (A E .) Then it follows that (D F) is parallel to (A P .) Therefore, in triangle (A S C) we have (C D / S D=C F / F A=3 / 2 .) But (D C=B D / 3=) ((B S+S D) / 3 .) Therefore (B S / S D=7 / 2)
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