In the parallelogram ABCD, am is perpendicular to BC and an is perpendicular to CD. It is proved that am: ab = Mn: AC
Proof: let ∠ B = a
⊙ parallelogram 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
And ∵ ad = BC
∴ AM / CD=AN / AD
In △ amn and △ DCA, am / CD = an / AD, ∠ man = ∠ D
∴△AMN∽△DCA
∴MN / CA = AM / DC
And ∵ AB = CD
It is proved that Mn / Ca = am / AB, i.e. am: ab = Mn: AC
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