Ta có: $AM//BN$ theo định lý Ta-let $\Delta CBN$ có:
$\dfrac{AM}{NB}=\dfrac{CM}{CB}$
$AM//PC$ theo định lý Ta-let $\Delta BCP$ có:
$\dfrac{AM}{PC}=\dfrac{BM}{BC}$
$\Rightarrow AM\left({\dfrac1{BN} +\dfrac 1{CP}}\right) =\dfrac{ AM}{BN} +\dfrac{ AM}{CP}$
$= \dfrac{CM}{BC }+\dfrac {BM}{BC} =\dfrac{ CM + BM}{BC} =\dfrac{ BC}{BC} = 1$
$⇔ \dfrac1{BN} +\dfrac 1{CP} =\dfrac 1{AM}$ (đpcm)