# 梅涅劳斯定理

${\displaystyle {\frac {AN}{NB}}\cdot {\frac {BL}{LC}}\cdot {\frac {CM}{MA}}=1}$

${\displaystyle {\frac {AN}{NB}}\cdot {\frac {BL}{LC}}\cdot {\frac {CM}{MA}}=1}$

LMN三点共线。利用这个逆定理，可以判断三点共线。 如果在上式中线段用有向线段表示，那么右面的结果为-1。

## 证明

#### 面积法证明

${\displaystyle {\frac {AN}{NB}}\cdot {\frac {BL}{LC}}\cdot {\frac {CM}{MA}}={\frac {\mathrm {S} _{\triangle LNA}}{\mathrm {S} _{\triangle LNB}}}\cdot {\frac {\mathrm {S} _{\triangle LNB}}{\mathrm {S} _{\triangle LNC}}}\cdot {\frac {\mathrm {S} _{\triangle LNC}}{\mathrm {S} _{\triangle ANL}}}=1}$

#### 正弦定理证明

${\displaystyle {\frac {AN}{AM}}={\frac {\sin \beta }{\sin \alpha }},}$

${\displaystyle {\frac {BL}{BN}}={\frac {\sin \alpha }{\sin \gamma }},}$

${\displaystyle {\frac {CM}{CL}}={\frac {\sin \gamma }{\sin \beta }}.}$

${\displaystyle {\frac {AN}{AM}}\cdot {\frac {BL}{BN}}\cdot {\frac {CM}{CL}}={\frac {\sin \beta }{\sin \alpha }}\cdot {\frac {\sin \alpha }{\sin \gamma }}\cdot {\frac {\sin \gamma }{\sin \beta }}=1,}$

${\displaystyle {\frac {AN}{NB}}\cdot {\frac {BL}{LC}}\cdot {\frac {CM}{MA}}=1.}$

## 延伸阅读

• Russell, John Wellesley. Ch. 1 §6 "Menelaus' Theorem". Pure Geometry. Clarendon Press. 1905.

## 参考文献

1. ^ Smith, D.E. History of Mathematics II. Courier Dover Publications. 1958: 607. ISBN 0-486-20430-8.
2. Rashed, Roshdi. Encyclopedia of the history of Arabic science 2. London: Routledge. 1996: 483. ISBN 0-415-02063-8.
3. Moussa, Ali. Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations. Arabic Sciences and Philosophy (Cambridge University Press). 2011, 21 (1). doi:10.1017/S095742391000007X.