ratio chasing inside a triangle
Source: Dutch NMO 1997 p2
January 29, 2020
ratiogeometry
Problem Statement
The lines and intersect in within a triangle .
It is given that and . Determine the ratio .[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("", A, SW);
dot("", B, SE);
dot("", C, N);
dot("", D, NE);
dot("", E, W);
dot("", F, dir(270));
dot("", S, NE);
[/asy]