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Routh's Theorem

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In triangle ABC, D, E and F are points on sides BC, AC, and AB, respectively. Let r=\frac{AF}{AB}, s=\frac{BD}{BC}, and =\frac{CE}{CA}. Let G be the intersection of AD and BC, H be the intersection of BE and CF, and I be the intersection of CF and AD. Then, Routh's Theorem states that

[GHI]=\dfrac{(rst-1)^2}{(rs+r+1)(st+s+1)(tr+t+1)}[ABC]

unitsize(5); defaultpen(fontsize(10));pair A,B,C,D,E,F,G,H,I;A=(10,20);B=(0,0);C=(30,0);D=(20,0);E=(16.66,13.33);F=(5,10);G=(...

Proof

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