Giải thích các bước giải:
a.Ta có :
$AD//BC\to\widehat{BMA}=\widehat{DAN}$
$AB//DN\to \widehat{BAM}=\widehat{AND}$
$\to\Delta ABM\sim\Delta NDA(g.g)$
$\to\dfrac{BM}{DA}=\dfrac{AB}{DN}\to BM.DN=AB.AD$ không đổi
b.Ta có :
$AD//BM\to\dfrac{AP}{PM}=\dfrac{PD}{PB}$
$\to\dfrac{AP}{AP+PM}=\dfrac{PD}{PD+PB}$
$\to\dfrac{AP}{AM}=\dfrac{PD}{BD}$
Lại có :
$AB//DN\to\dfrac{PA}{PN}=\dfrac{PB}{PD}$
$\to\dfrac{PA}{PA+PN}=\dfrac{PB}{PD+PB}$
$\to\dfrac{AP}{AN}=\dfrac{PB}{BD}$
$\to\dfrac{AP}{AM}+\dfrac{AP}{AN}=\dfrac{PD}{BD}+\dfrac{PB}{BD}$
$\to\dfrac{AP}{AM}+\dfrac{AP}{AN}=\dfrac{PD+PB}{BD}$
$\to\dfrac{AP}{AM}+\dfrac{AP}{AN}=\dfrac{BD}{BD}$
$\to\dfrac{AP}{AM}+\dfrac{AP}{AN}=1$
$\to\dfrac{1}{AM}+\dfrac1{AN}=\dfrac1{AP}$
