# Routh's Theorem

Routh's Theorem

Suppose that in a triangle lines are drawn from the vertices to the opposite sides at points , , and . Let , , and denote the ratios , , and . Let the triangle formed inside be .

ABC

U

V

W

λ

μ

ν

AV

VC

CU

UB

BW

WA

ABC

RST

Routh's theorem states that .

area[RST]=area[ABC]

(λμν-1)

2

((μν+μ+1)(νλ+ν+1)(λμ+λ+1))