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2

Part of 1997 Dutch Mathematical Olympiad

Problems(1)

ratio chasing inside a triangle

Source: Dutch NMO 1997 p2

1/29/2020
The lines AD,BEAD , BE and CFCF intersect in SS within a triangle ABCABC . It is given that AS:DS=3:2AS: DS = 3: 2 and BS:ES=4:3BS: ES = 4: 3 . Determine the ratio CS:FSCS: FS .
[asy] unitsize (1 cm);
pair A, B, C, D, E, F, S;
A = (0,0); B = (5,0); C = (1,4); S = (14*A + 15*B + 6*C)/35; D = extension(A,S,B,C); E = extension(B,S,C,A); F = extension(C,S,A,B);
draw(A--B--C--cycle); draw(A--D); draw(B--E); draw(C--F);
dot("AA", A, SW); dot("BB", B, SE); dot("CC", C, N); dot("DD", D, NE); dot("EE", E, W); dot("FF", F, dir(270)); dot("SS", S, NE); [/asy]
ratiogeometry