a/ \(AB=3AM↔\dfrac{AB}{AM}=3\)

\(AC=3AN↔\dfrac{AC}{AN}=3\)

 \(→\dfrac{AB}{AM}=\dfrac{AC}{AN}=3\)

Xét \(ΔABC\) và \(ΔAMN\):

\(\dfrac{AB}{AM}=\dfrac{AC}{AN}(cmt)\)

\(\widehat{A}:chung\)

\(→ΔABC\backsim ΔAMN(c-g-c)\)

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b/ \(\dfrac{AB}{AM}=\dfrac{AN}{AN}\\→MN//BC\\→\begin{cases}\widehat{KMN}=\widehat{KCB}\\\widehat{KNM}=\widehat{KBC}\end{cases}\)

Xét \(ΔKMN\) và \(ΔKCB\):

\(\widehat{KMN}=\widehat{KCB}(cmt)\)

\(\widehat{KNM}=\widehat{KBC}(cmt)\)

\(→ΔKMN\backsim ΔKCB(g-g)\)

\(→\dfrac{KM}{KN}=\dfrac{KC}{KB}\)

\(↔KM.KB=KN.KC\)

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c/ \(ΔABC\backsim ΔAMN\)

\(→\dfrac{AB}{AM}=\dfrac{BC}{MN}=3\)

\(ΔKMN\backsim ΔKCB\)

\(→\dfrac{BC}{NM}=\dfrac{KM}{KC}=\dfrac{KN}{KB}=3=k\)

\(\dfrac{S_{ΔKBC}}{S_{ΔKMN}}=k^2\)

\(→\dfrac{S_{ΔKBC}}{5}=3^2=9\)

\(→S_{ΔKBC}=45(cm^2)\)