证：设∠B= a
∵平行四边形ABCD,AM⊥BC,AN⊥CD
∴∠B=∠D=a,∠BAD=180°-a,∠BAM=∠DAN=90°-a
S□ABCD=AM × BC= AN × CD,AB=CD,AD=BC
∴∠MAN=∠BAD- ∠BAM-∠DAN=a=∠D
AM / CD=AN / BC
又∵AD=BC
∴ AM / CD=AN / AD
∴在△AMN和△DCA 中,AM / CD=AN / AD,∠MAN=∠D
∴△AMN∽△DCA
∴MN / CA = AM / DC
又∵AB=CD
∴MN / CA = AM / AB 即 AM：AB=MN：AC,得证