共边定理

  在一些特殊情况下,共边三角形的面积比可以转化成线段比表示.用数学语言来表示就是共边定理

• 设直线AB与PQ交于M,则
  • ${\displaystyle {\frac {\triangle {PAB}}{\triangle {QAB}}}}$ =${\displaystyle {\frac {PM}{QM}}}$ 
• 证明:有4种情形:;P,Q在直线AB两侧,A,B也在直线PQ两侧

P,Q在直线AB同侧,A,B在直线PQ两侧

PQ在直线AB两侧,A,B在直线PQ同侧

P,Q在直线AB两侧,A,B也在直线PQ同侧

• • ${\displaystyle {\frac {\triangle {PAB}}{\triangle {QAB}}}}$ =${\displaystyle {\frac {\triangle {PAB}}{\triangle {PMB}}}\cdot {\frac {\triangle {PMB}}{\triangle {QMB}}}\cdot {\frac {\triangle {QMB}}{\triangle {QAB}}}}$ =${\displaystyle {\frac {AB}{MB}}\cdot {\frac {PM}{QM}}\cdot {\frac {MB}{AB}}}$ =${\displaystyle {\frac {PM}{QM}}}$ 
• 证明2:在AB上取一点N使MN=AB,立刻得到:

${\displaystyle {\frac {\triangle {PAB}}{\triangle {QAB}}}}$ =${\displaystyle {\frac {\triangle {PMN}}{\triangle {QMN}}}}$ =${\displaystyle {\frac {PM}{QM}}}$