Vì \(\frac{MA}{MB}=\frac25\)
nên \(S_{CMA}=\frac25\times S_{CMB};S_{IMA}=\frac25\times S_{IMB}\)
=>\(S_{CMA}-S_{IMA}=\frac25\times\left(S_{CMB}-S_{IMB}\right)\)
=>\(S_{CIA}=\frac25\times S_{CIB}\)
Vì \(\frac{BN}{NC}=\frac13\)
nên \(S_{ANB}=\frac13\times S_{ANC};S_{INB}=\frac13\times S_{INC}\)
=>\(S_{ANB}-S_{INB}=\frac13\times\left(S_{ANC}-S_{INC}\right)\)
=>\(S_{AIB}=\frac13\times S_{AIC}\)
=>\(S_{AIB}=\frac13\times\frac25\times S_{CIB}=\frac{2}{15}\times S_{CIB}\)
Vì \(\frac{BN}{NC}=\frac13\)
nên NC=3BN
Ta có: NC+BN=BC
=>BC=3BN+BN=4BN
=>\(BN=\frac14\cdot BC\)
=>\(S_{INB}=\frac14\cdot S_{IBC}\)
=>\(\frac{S_{AIB}}{S_{IBN}}=\frac{2}{15}:\frac14=\frac{8}{15}\)
=>\(\frac{AI}{IN}=\frac{8}{15}\)
=>\(\frac{AI}{AN}=\frac{8}{23}\)
Vì \(\frac{AM}{MB}=\frac25\)
nên \(AM=\frac25MB\)
Ta có: AM+MB=AB
=>\(AB=\frac25MB+MB=\frac75MB\)
=>\(MB=\frac57BA\)
=>\(S_{BMI}=\frac57\cdot S_{BIA}=\frac57\cdot\frac{2}{15}\cdot S_{BIC}=\frac{10}{105}\cdot S_{BIC}=\frac{2}{21}\cdot S_{BIC}\)
=>\(\frac{MI}{IC}=\frac{2}{21}\)
=>\(\frac{CI}{IM}=\frac{21}{2}\)
