a: Ta có: AM+MB=AB

=>\(MB=AB-\frac13\times AB=\frac23\times AB\)

=>\(S_{BMC}=\frac23\times S_{ABC}\) (1)

Ta có: AN+NC=AC
=>\(NC=AC-AN=AC-\frac13\times AC=\frac23\times AC\)

=>\(S_{BNC}=\frac23\times S_{ABC}\) (2)

Từ (1),(2) suy ra \(S_{BMC}=S_{BNC}\)

b: Ta có: \(AN=\frac13\times AC\)

=>\(S_{ABN}=\frac13\times S_{ABC}\)

Ta có: \(AM=\frac13\times AB\)

=>\(S_{AMN}=\frac13\times S_{ANB}=\frac13\times\frac13\times S_{ABC}=\frac19\times S_{ABC}=\frac{180}{9}=20\left(\operatorname{cm}^2\right)\)

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a: \(AM=\frac23MC\)

=>\(MC=\frac32AM\)

AM+MC=AC

=>\(AC=AM+\frac32AM=\frac52AM\)

=>\(AM=\frac25AC\)

=>\(S_{ABM}=\frac25\times S_{ABC}=\frac25\times60=24\left(\operatorname{cm}^2\right)\)

b: Ta có: BN+NC=BC

=>\(BC=BN+\frac13BN=\frac43BN\)

=>\(CN=\frac14CB;BN=\frac34BC\)

TA có: \(CM=\frac35\times CA\)

=>\(S_{BMC}=\frac35\times S_{ABC}\)

\(CN=\frac14CB\)

=>\(S_{MNC}=\frac14\times S_{BMC}=\frac14\times\frac35\times S_{ABC}=\frac{3}{20}\times S_{ABC}\)

TA có: \(S_{AMNB}+S_{MNC}=S_{ABC}\)

=>\(S_{AMNB}=S_{ABC}-\frac{3}{20}\times S_{ABC}=\frac{17}{20}\times S_{ABC}=\frac{17}{20}\times60=51\left(\operatorname{cm}^2\right)\)

c: Ta có: \(AM=\frac23\times MC\)

=>\(S_{BMA}=\frac23\times S_{BMC};S_{OMA}=\frac23\times S_{OMC}\)

=>\(S_{BMA}-S_{OMA}=\frac23\times\left(S_{BMC}-S_{OMC}\right)\)

=>\(S_{BOA}=\frac23\times S_{BOC}\)

=>\(S_{BOC}=\frac32\times S_{BOA}\)

Ta có: \(NC=\frac13\times NB\)

=>\(S_{ANC}=\frac13\times S_{ANB};S_{ONC}=\frac13\times S_{ONB}\)

=>\(S_{ANC}-S_{ONC}=\frac13\times\left(S_{ANB}-S_{ONB}\right)\)

=>\(S_{AOC}=\frac13\times S_{AOB}=\frac29\times S_{BOC}\)

Ta có: \(S_{BOC}+S_{AOC}+S_{AOB}=S_{ABC}\)

=>\(S_{ABC}=S_{OAB}+\frac32\times S_{OAB}+\frac13\times S_{OAB}=S_{OAB}\left(\frac43+\frac32\right)=S_{AOB}\cdot\frac{11}{6}\)

=>\(S_{AOB}=\frac{6}{11}\cdot S_{ABC}=\frac{6}{11}\cdot60=\frac{360}{11}\left(\operatorname{cm}^2\right)\)

=>\(\frac{S_{AOB}}{S_{ABN}}=\frac{\frac{6}{11}\cdot S_{ABC}}{\frac34\cdot S_{ABC}}=\frac{6}{11}:\frac34=\frac{6}{11}\times\frac43=\frac{24}{33}=\frac{8}{11}\)

=>\(\frac{AO}{AN}=\frac{8}{11}\)

=>\(\frac{AO}{ON}=\frac83\)