a,
$\Delta$ AMB và $\Delta$ ANC có:
$\widehat{AMH}= \widehat{ANC}$
AM= AN
MB= NC
=> $\Delta$ AMB= $\Delta$ ANC (c.g.c)
=> AB= AC
=> $\Delta$ ABC cân A
b,
$\Delta$ ABC cân A nên $\widehat{ABC}= \widehat{ACB}$
=> $\widehat{MBH}= \widehat{NCK}$ (đối đỉnh)
$\Delta$ MBH và $\Delta$ NCK có:
$\widehat{MHB}= \widehat{NKC}= 90^o$
MB= NC
$\widehat{MBH}= \widehat{NCK}$
=> $\Delta$ MBH= $\Delta$ NCK (ch.gn) (*)
c,
(*) => $\widehat{BMH}= \widehat{CNK}$
=> $\Delta$ OMN cân tại O
d,
Khi $\widehat{BAC}= 60^o$, $\Delta$ ABC đều.
=> $\widehat{ABC}= \widehat{ACB}= 60^o$
=> $\widehat{ABM}= \widehat{ACN}= 120^o$ (kề bù)
Mà MB= BC= CN, AB= BC= AC
=> MB= AB, NC= AC
=> $\Delta$ ABM cân B, $\Delta$ ACN cân C
=> $\widehat{AMB}= \widehat{ANC}= \frac{180-120}{2}=30^o$
=> $\widehat{MAN}= \widehat{MAB}+ \widehat{BAC}+ \widehat{CAN}= 30+60+30= 120^o$
