羅斯定理

定理

${\displaystyle {\frac {(xyz-1)^{2}}{(xy+y+1)(yz+z+1)(zx+x+1)}}.}$

證明

${\displaystyle {\frac {AF}{FB}}\times {\frac {BC}{CD}}\times {\frac {DR}{RA}}=1}$

${\displaystyle {\frac {DR}{RA}}={\frac {BF}{FA}}{\frac {DC}{CB}}={\frac {zx}{x+1}}}$

${\displaystyle S_{ARC}={\frac {AR}{AD}}S_{ADC}={\frac {AR}{AD}}{\frac {DC}{BC}}S_{ABC}={\frac {x}{zx+x+1}}}$

${\displaystyle \displaystyle S_{PQR}=S_{ABC}-S_{ARC}-S_{BPA}-S_{CQB}}$
${\displaystyle =1-{\frac {x}{zx+x+1}}-{\frac {y}{xy+y+1}}-{\frac {z}{yz+z+1}}}$
${\displaystyle ={\frac {(xyz-1)^{2}}{(xz+x+1)(yx+y+1)(zy+z+1)}}.}$

參攷

• Murray S. Klamkin and A. Liu (1981) "Three more proofs of Routh's theorem", Crux Mathematicorum 7:199–203.
• H. S. M. Coxeter (1969) Introduction to Geometry, statement p. 211, proof pp. 219–20, 2nd edition, Wiley, New York.
• J. S. Kline and D. Velleman (1995) "Yet another proof of Routh's theorem" (1995) Crux Mathematicorum 21:37–40
• Routh's Theorem, Jay Warendorff, The Wolfram Demonstrations Project.
• （英文）
• Routh's Theorem by Cross Products at MathPages
• Ayoub, Ayoub B. (2011/2012) "Routh's theorem revisited", Mathematical Spectrum 44 (1): 24-27.