View source for Routh's Theorem

In [[triangle]] <math>ABC</math>, <math>D</math>, <math>E</math> and <math>F</math> are points on sides <math>BC</math>, <math>AC</math>, and <math>AB</math>, respectively. Let <math>r=\frac{AF}{FB}</math>, <math>s=\frac{BD}{DC}</math>, and <math>t=\frac{CE}{AE}</math>. Let <math>G</math> be the intersection of <math>AD</math> and <math>BC</math>, <math>H</math> be the intersection of <math>BE</math> and <math>CF</math>, and <math>I</math> be the intersection of <math>CF</math> and <math>AD</math>. Then, '''Routh's Theorem''' states that 

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

<asy>
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=(14.585,11.6298);
H=(9.998,8);
I=(17.5,5);
draw(A--B);
draw(B--C);
draw(C--A);
draw(A--D);
draw(B--E);
draw(C--F);
label("$A$",A,N);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,S);
label("$E$",E,NE);
label("$F$",F,NW);
label("$G$",G,N);
label("$H$",H,N);
label("$I$",I,SW);</asy>

==Proof==
{{incomplete|proof}}

{{stub}}
== See also ==
* [[Menelaus' Theorem]]
*[[Ceva's Theorem]]


[[Category:Geometry]]

[[Category:Theorems]]
[[Category:Geometry]]
[[Category:Definition]]